Differential Calculus is a branch of Calculus in mathematics that deals with the study of the rates at which quantities change. It involves calculating derivatives and using them to solve problems involving nonconstant rates of change. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications.
In this article, we have tried to provide a brief overview of the branch of Differential Calculus including topics such as limits, derivatives, various formulas for derivatives as well as application of derivatives.
What is Differential Calculus?
Differential calculus is a branch of calculus that deals with the study of rates of change of functions and the behavior of these functions in response to infinitesimal changes in their independent variables.
In differential calculus, we study derivatives, differentiation techniques (Power, Product, Quotient, Chain rules), implicit differentiation, higherorder derivatives, applications (optimization, related rates, curve sketching), tangent lines, critical points, extrema (max/min values), and many more.
Important Terms in Differential Calculus
Some of the most common terms used in Differential Calculus are:
 Independent Variable
In calculus, the independent variable is the variable whose value doesn’t depend on any other variable. For example, in the equation f(x) = 2x^{2}, x is the independent variable.
 Dependent Variable
In calculus, the dependent variable is a variable whose value depends on another variable, i.e., the independent variable. For example, in the equation y = f(x) = 2x^{2}, y is the dependent variable, as its value can only be determined for some chosen value of x.
 Function
In calculus, a function is a mathematical rule or relationship that assigns exactly one output value to each input value. This relationship is denoted symbolically as y = f(x), where x is the independent variable and y is the dependent variable.
What is Limit?
For a function y = f(x), then limit x approaches a for function y = f(x) represents the value function approaches when we approach the input value x = a. In simple words, the limit of any function at a given point tells us about its behaviour at and around the point of consideration. It is given as lim _{x⇝a} f(x). Limit is unique in nature i.e. for x tends to a, there can’t be two values of f(x).
Left Hand and Right Hand Limit
LeftHand Limit  [Tex]\lim_{x \to a^{}}f(x)=\lim_{h \to 0^{}}f(ah)[/Tex] 

RightHand Limit  [Tex]\lim_{x \to a^{+}}f(x)=\lim_{h \to 0^{+}}f(a+h)[/Tex] 
Existence of Limit
For Existence of [Tex]\bold{\lim_{x \to a}f(x)} [/Tex],
 Both [Tex]\lim_{x \to a^{}}f(x) [/Tex]and [Tex]\lim_{x \to a^{+}}f(x) [/Tex]exists, and
 [Tex]\lim_{x \to a^{}}f(x) = \lim_{x \to a^{+}}f(x)[/Tex]
Properties of Limits
If there are two functions f(x) and g(x) such that their limits [Tex]\lim_{x \to a}f(x) [/Tex]and [Tex]\lim_{x \to a}g(x) [/Tex]exist then following properties are followed
[Tex]\lim _{x \rightarrow a}[f(x) \pm g(x)][/Tex]  [Tex]\lim _{x \rightarrow a} f(x) \pm \lim _{x \rightarrow a} g(x)[/Tex] 
[Tex]\lim _{x \rightarrow a}[k f(x)][/Tex]  [Tex]k \lim _{x \rightarrow a} f(x),[/Tex] 
[Tex]\lim _{x \rightarrow a}[f(x) g(x)][/Tex]  [Tex]\lim _{x \rightarrow a} f(x) \lim _{x \rightarrow a} g(x)[/Tex] 
[Tex]\lim _{x \rightarrow a} \frac{f(x)}{g(x)}[/Tex]  [Tex]\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}, [/Tex]provided [Tex]\lim _{x \rightarrow a} g(x) \neq 0[/Tex] 
[Tex]\lim _{x \rightarrow a}[f(x)]^{g(x)}[/Tex]  [Tex]\left[\lim _{x \rightarrow a} f(x)\right]^{\lim _{x \rightarrow a} g(x)}[/Tex] 
[Tex]\lim _{x \rightarrow a}(g \circ f)(x)[/Tex]  [Tex]\lim _{x \rightarrow a} g[f(x)]=g\left[\lim _{x \rightarrow a} f(x)\right][/Tex] 
[Tex]\lim _{x \rightarrow a} \log f(x)[/Tex]  [Tex]\log \left[\lim _{x \rightarrow a} f(x)\right], [/Tex]provided [Tex]\lim _{x \rightarrow a} f(x)>0.[/Tex] 
[Tex]\lim _{x \rightarrow a} e^{f(x)}[/Tex]  [Tex]e^{\lim _{x \rightarrow a} f(x)}[/Tex] 
[Tex]\lim _{x \rightarrow a}f(x)[/Tex]  [Tex]\left\lim _{x \rightarrow a} f(x)\right[/Tex] 
[Tex]\lim _{x \rightarrow a} f(x)=+\infty \ or \infty[/Tex]  [Tex]\lim _{x \rightarrow a} \frac{1}{f(x)}=0[/Tex] 
Limit Formulas
Some of the common formulas for limits are:
[Tex]\lim _{x \rightarrow a} \frac{x^na^n}{xa}[/Tex]  [Tex]n a^{n1}, n \in Q[/Tex] 
[Tex]\lim _{x \rightarrow 0} \frac{(1+x)^n1}{x}[/Tex]  [Tex]n, n \in Q[/Tex] 
[Tex]\lim _{x \rightarrow 0} \frac{\sin x}{x}=\lim _{x \rightarrow 0} \frac{x}{\sin x}[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{\tan x}{x}=\lim _{x \rightarrow 0} \frac{x}{\tan x}[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{\sin ^{1} x}{x}=\lim _{x \rightarrow 0} \frac{x}{\sin ^{1} x}[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{\tan ^{1} x}{x}=\lim _{x \rightarrow 0} \frac{x}{\tan ^{1} x}[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{\sin x^{\circ}}{x}[/Tex]  [Tex]\frac{\pi}{180}[/Tex] 
[Tex]\lim _{x \rightarrow 0} \cos x[/Tex]  1 
[Tex]\lim _{x \rightarrow a} \frac{\sin (xa)}{xa}[/Tex]  1 
[Tex]\lim _{x \rightarrow a} \frac{\tan (xa)}{xa}[/Tex]  1 
[Tex]\lim _{x \rightarrow a} \sin ^{1} x[/Tex]  [Tex]\sin ^{1} a,a \leq 1[/Tex] 
[Tex]\lim _{x \rightarrow a} \cos ^{1} x[/Tex]  [Tex]\cos ^{1} a,a \leq 1[/Tex] 
[Tex]\lim _{x \rightarrow a} \tan ^{1} x[/Tex]  [Tex]\tan ^{1} a,\infty<a<\infty[/Tex] 
[Tex]\lim _{x \rightarrow \infty} \frac{\sin x}{x}=\lim _{x \rightarrow \infty} \frac{\cos x}{x}[/Tex]  0 
[Tex]\lim _{x \rightarrow \infty} \frac{\sin \frac{1}{x}}{\frac{1}{x}}[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{1\cos x}{x}[/Tex]  0 
[Tex]\lim _{x \rightarrow 0} \frac{e^x1}{x}[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{a^x1}{x}[/Tex]  [Tex]\log _e a, a>0[/Tex] 
[Tex]\lim _{x \rightarrow 0} \frac{\log _e(1+x)}{x}[/Tex]  1 
[Tex]\lim _{x \rightarrow e} \log _e x[/Tex]  1 
[Tex]\lim _{x \rightarrow 0} \frac{\log _a(1+x)}{x}[/Tex]  [Tex]\log _a e, a>0, \neq 1[/Tex] 
Learn More Limit Formulas.
Evaluation of Limits
Limits can be solved with different methods depending on the type of form it exhibits for x = a.
 Determinate Forms
 Indeterminate Forms
Determinate Forms
If at x = a, f(x) yields a definite value then the limit is calculated by \lim_{x \to a}f(x)=f(a).
Indeterminate Forms
If at x = a, f(x) yields a value in the form of 0/0, ∞/∞, ∞∞, 0^{0},1^{∞}, and ∞^{0} then it can be solved by following mentioned methods:
 Factorization Method
It is used when [Tex]\lim_{x \to a}\frac{f(x)}{g(x)} [/Tex]takes the form of 0/0 then xa is a factor of the numerator and denominator which can be cancelled to make it into determinate form and then solve.
 Rationalization Method
This method is used when [Tex]\lim_{x \to a}\frac{f(x)}{g(x)} [/Tex]takes the form of 0/0 or ∞/∞ and the denominator is in square root form. In this case, the denominator is rationalized.
 Substitution Method
In this case, the x in f(x) is replaced with x = a + h or a – h such that when x tends to a then h tends to 0.
When x→∞: In this case when [Tex]\lim_{x \to ∞}\frac{f(x)}{g(x)} [/Tex]takes the form of ∞/∞ then the numerator and denominator are divided by the highest power of x.
Learn More, Strategy in Finding Limits
L Hospital Rule
L Hospital Rule states that if f(x)/g(x) is in the form of 0/0 or ∞/∞ for x = a then \lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)} , where f'(x) and g'(x) are the first order derivatives of functions f(x) and g(x) respectively.
Sandwich Theorem
Sandwich Theorem states that for given functions f(x), g(x), and h(x) that exists in the order f(x) ≤ g(x) ≤ h(x) for x belonging to the common domain then for some value ‘a’ if [Tex] \bold{\lim_{x \to a}f(x)} = p = \bold{\lim_{x \to a}h(x)} [/Tex] then \bold{\lim_{x \to a}g(x)} = p
Continuity, Discontinuity, and Differentiability of a Function
The conditions for continuity, discontinuity, and differentiability of a function at a point are tabulated below:
Continuity  Discontinuity  Differentiability 

[Tex]\lim_{x \to a^{}}f(x)=\lim_{x \to a^{+}}f(x)=f(a)[/Tex] 


Fundamental Theorems of Continuity
 If f and g are continuous functions then f±g, fg, f/g (where, g≠0), and cf(x) all are continuous.
 If g is continuous at ‘a’ and f is continuous at g(a) then fog is also continuous at ‘a’.
 If f is continuous in its domain then f is also continuous in its domain
 If f is continuous in the domain D then 1/f is also continuous in D{x:f(x) = 0}.
Fundamental Theorems of Differentiability
 If f and g are two differential functions then their sum, difference, product, and quotients are also differentiable.
 If f and g are two differential functions then fog is also differentiable.
 If f and g are not differential functions then their sum and product functions can be differential functions.
Learn More Continuity and Discontinuity
Derivatives
Derivative is defined as the change in the output of a function with respect to the given input. This change is used to analyze the various physical factors associated with the function. Now we will look at the basic expression of Derivatives
Derivative at a Point  [Tex]\bold{\lim_{x \to c}\frac{f(x)f(c)}{xc}}[/Tex] 

Derivative of a Function  [Tex]\bold{f'(x) = \lim_{h \to 0} \frac{f(x + h) f(x)}{h}}[/Tex] 
Derivative as a Rate Measure  [Tex]\dfrac{dy}{dx}=\bold{\lim_{\Delta x \to 0}\dfrac{\Delta y}{\Delta x}}[/Tex] 
Differentiation from the First Principle  [Tex]\bold{f'(x) = \lim_{h \to 0} \frac{f(x + h) f(x)}{h}}[/Tex] 
Algebra of Derivatives
In this section, we will learn how to find the derivative of two functions given in the form of an algebraic expression.
Sum Rule  {f(x) + g(x)}’ = f'(x) + g'(x) 

Difference Rule  {f(x) – g(x)}’ = f'(x) – g'(x) 
Product Rule  {f(x).g(x)}’ = f'(x).g(x) + f(x).g'(x) 
Quotient Rule  {f(x)/g(x)}’ = {f'(x).g(x) – f(x).g'(x)}/(g(x))^{2} 
Learn More, Algebra of Derivatives
Differentiation Formulas
Some of the most common formula used to find derivative are tabulated below:
d/dx(c)  0 
d/dx{c.f(x)}  c.f'(x) 
d/dx(x)  1 
d/dx(x^{n})  nx^{n1} 
d/dx{f(g(x))}  f'(g(x)).g'(x) 
d/dx(a^{x})  a^{x}.ln(a) 
d/dx{ln(x)} {Note: ln(x) = log_{e}(x)}  1/x, x>0 
d/dx(log_{a}x)  1/xln(a) 
d/dx(e^{x})  e^{x} 
d/dx{sin(x)}  cos(x) 
d/dx{cos(x)}  sin(x) 
d/dx{tan(x)}  sec^{2}x 
d/dx{sec(x)}  sec(x).tan(x) 
d/dx{cosec(x)}  cosec(x).cot(x) 
d/dx{cot(x)}  cosec^{2}(x) 
d/dx{sin^{1}(x)}  1/√(1 – x^{2}) 
d/dx{cos^{1}(x)}  1/√(1 – x^{2}) 
d/dx{tan^{1}(x)}  1/(1+x^{2}) 
Check,
Differentiation of Function of a Function
It says that if f(x) and g(x) are differentiable functions then fog is also differentiable.
d/dx{(fog)(x)} = d/dx{fog(x)}.d/dx{g(x)}
OR
y = g(x) and z = f(y), then dz/dx = (dz/dy).(dy/dx)
Chain Rule
When we need to differentiate the function of a function, we apply the chain rule. In Chain Rule, we first differentiate the first function and then differentiate the second function and write their derivatives in product form. Some of the examples are mentioned below:
d/dx[{f(x)}^{n}]  n[{f(x)}^{n1}].f'(x) 
d/dx[ln{f(x)}]  {1/f(x)}.f'(x) 
d/dx{e^{f(x)}}  e^{f(x)}.f'(x) 
d/dx{sin(f(x))}  cos(f(x)).f'(x) 
d/dx{cos(f(x))}  sin(f(x)).f'(x) 
d/dx{tan(f(x))}  sec^{2}(f(x)).f'(x) 
d/dx{tan^{1}(f(x))} = {1/(1+(f(x))2)}.f'(x)  {1/(1+(f(x))^{2})}.f'(x) 
Differentiation of a Function with Respect to Another Function
Let’s say we have two functions, u = f(x) and v = g(x) then, differentiation of u with respect to v is found in the following manner
Step 1: Find du/dx
Step 2: Find dv/dx
Step 3: du/dv = (du/dx)/(dv/dx)
Differentiation of Determinant
Let’s say we have to differentiate a determinant given in terms of x, then its derivative is given as differentiation of row (or column) at a time.
If \bold{x = \begin{vmatrix}f(x)&g(x) \\ u(x)&v(x)\end{vmatrix}}
Then [Tex]\bold{x’ = \begin{vmatrix}f'(x)&g'(x) \\ u(x)&v(x)\end{vmatrix} + \begin{vmatrix}f(x)&g(x) \\ u'(x)&v'(x)\end{vmatrix}}[/Tex]
Where x’ represents the first order derivative fo x.
Implicit Differentiation
Implicit Differentiation is used when a function is not defined explicitly in terms of only one independent variable. In this case, the function is given as g(x,y). It should be noted that here y is equal to f(x). Hence, the differentiation is done in the following manner:
Let’s say we have a function g(x,y) = x^{2} + y + 3xy
then g'(x,y) = d{g(x,y)}/dx = 2x + y’ + 3y + 3xy’
⇒ g'(x,y) = 2x + 3y + y'(1 + 3x).
In implicit differentiation, the chain rule is used, and also product or quotient rule is used wherever applicable. For Example, in the case of 3xy, the product rule is used as x and y are in a product form which gives differentiation of 3xy as 3y + 3xy’.
Higher Order Derivatives
Higher Order Derivatives refer to the derivative of derivative of a function. In this, we first differentiate a function and find its derivatives and then again differentiate the derivative obtained for the first time. If differentiation is done two times then it is called Second Order Derivative and if done for ‘n’ times it is called n^{th} order derivative. For a function defined as y = f(x), its higherorder derivatives are given as follows:
Second Order Derivative  f”(x) = f^{2}(x) = d^{2}y/dx^{2} = {f'(x)}’ 

n^{th} Order Derivative  f^{n}(x) = d^{n}y/dx^{n} = {f^{(n1)}(x)}’ 
Error
Differential Error is a method used to calculate error in output for a change in input. To calculate the differential error, follow the belowmentioned steps
Step 1: Take x as the independent variable and x + Δx as the change in the variable.
Step 2: Find Δx by (x + Δx) – x. Assume dx = Δx.
Step 3: Find dy/dx at the given value of x.
Step 4: dy = (dy/dx).dx
The dy obtained in step 4 gives the value of Δy.
Now find y for the given value of x as y = f(x). Thus, the change in y is given as y + Δy.
To find the percentage error multiply (Δy/y) with 100.
Approximation
Approximation is used to find the approximate value of nonperfect square roots or cube roots. To find the approximate value, use the following steps:
Step 1: Define the function. For example, if you have to find the approximate value of a square root take y = f(x) = √x.
Step 2: Take a value closer to the given value of x which is a perfect square or cube.
Step 3: Find the difference between the assumed value and the given value and take it as Δx = dx.
Step 4: Find (dy/dx) at the assumed value
Step 5: Now, dy = (dy/dx).dx
Step 6: dy = Δy is the change in y.
Hence, the approximate value is given as y + Δy.
Critical Point
Critical Point is the point where the derivative of the function is either zero or not defined. C is the critical point of the function f(x) if
(dy/dx)_{x= C} = 0
OR
(dy/dx)_{x= C} = Not Defined
Concave Up and Concave Down
Concave Up  f”(x) > 0 at x = a 

Concave Down  f”(x) < 0 at x = a 
Inflection Point
The point at which the concavity of a function changes is called the Inflection Point. The belowmentioned steps can be used to find the inflection point:
Step 1: Find the secondorder derivative of the function i.e. f”(x).
Step 2: Equate f”(x) = 0 and find x.
Step 3: Take a point smaller than x and one larger than x.
Step 4. Put the values in f”(x) and observe the sign. If the sign changes from a value lower than x to a value larger than x then an inflection point exists.
Tangent and Normal
For a curve defined by function f(x) and let us assume there is a Point P(x_{1},y_{1}) on it. Then,
Slope of the Tangent  (dy/dx)_{x = P} 

Slope of Normal  1/Slope of tangent at P = 1/(dy/dx)_{x = P} 
Learn More, Tangent and Normals
Increasing and Decreasing Function
Let f(x) is a function differentiable on (a,b) then the function is
Increasing  f'(x) > 0 

Decreasing  f'(x) < 0 
Constant  f'(x) = 0 
Learn More, Increasing and Decreasing Function
Maxima and Minima
Absolute Maximum  x = a  f(x) ≤ f(a) where x and a belong to domain of f. 

Absolute Minimum  x = a  f(x) ≥ f(a) where x and a belong to domain of f. 
Local Maxima  x = a  f(x) ≤ f(a) for all x near to a. 
Local Minima  x = a  f(x) ≥ f(a) for all x near to a. 
Learn More, Absolute Maxima and Minima and Relative Maxima and Minima
Fermat’s Theorem
If a function is differentiable at its local extremum then its derivative at that point must be zero i.e., if x = a is the local extrema of f(x) then f'(a) = 0.
Extreme Value Theorem
If f(x) is continuous in the closed interval [a,b] then there exists c ≥ a for which f(c) is minimum and d ≤ b for which f(d) is minimum. In short, for a ≤ c and d ≤ b, f(c) is the absolute minimum in the closed interval [a,b] and f(d) is the absolute maximum in the closed interval [a,b].
First Derivative Test
Local Minima  Local Maxima 

If x = a is point of local minima then
 If x = a is the point of local maxima then

Second Derivative Test
If x = a is a critical point of f(x) such that f'(a) = 0 then if
f”(a) < 0  x = a is the point of local maxima 

f”(a) > 0  x = a is the point of local minima 
f”(a) = 0  x = a may be a point of local maxima or local minima or none. 
Mean Value Theorem
Mean Value Theorem states that if a function f(x) is continuous in the closed interval [a,b] and differentiable in the open interval (a,b) then there exists a point c in (a,b) such that
f'(c) = [f(b) – f(a)]/(ba)
Learn More, Rolle’s Theorem and Lagrange’s Mean Value Theorem
Differential Equation
Differential Equation refers to an equation that has a dependent variable, an independent variable, and a differential coefficient of the dependent variable with respect to the independent variable.
Order and Degree of Differential Equation
Order of Differential Equation  Degree of Differential Equation 

Highest Derivative in the Equation Example: In (dy/dx)^{2} + 3(d^{2}y/dx^{2})^{3} Order is 2.  Exponent raised to Highest Derivative Example: In (dy/dx)^{3} + 3(d^{2}y/dx^{2})^{2} Degree is 3. 
Solution of First Order and First Degree Differential Equation
The solution of a firstorder and firstdegree differential equation can be found by different techniques depending upon the category they belong to.
Equation of the Standard Form
If a differential equation is in the form f[f1(x,y)]d{f1(x,y)} + Φ[f2(x,y)]d{f2(x,y)} + … = 0 then each term can be integrated separately. Some of the standard terms can be replaced by the exact differentials mentioned below:
xdy + ydx  d(xy) 
dx + dy  d(x + y) 
(xdy – ydx)/x^{2}  d(y/x) 
(ydx – xdy)/y^{2}  d(x/y) 
(xdy + ydx)/xy  d(log xy) 
(ydx – xdy)/xy  d(log x/y) 
(xdy – ydx)/xy  d(log y/x) 
(dx + dy)/x + y  d(log (x + y)) 
dx/x^{2} – dy/y^{2}  d(1/y – 1/x) 
Equation with Separable Variables
In the variable separable method, the equation is transformed into the form f(x)dx = g(y) and then integrated on both sides resulting in ∫f(x)dx = ∫g(y)dy + C.
Equation Reducible to Variable Separable Form
Let’s say we have a differential equation in the form dy/dx = f(ax + by + c) then
assume ax + by + c = p
Now differentiate both sides which results in a + b(dy/dx) = dp/dx
Hence, dy/dx = 1/b(dp/dx – a).
Now we already have dy/dx = f(ax + by + c) = f(p)
⇒ 1/b(dp/dx – a) = f(p)
⇒ dp/dx = b.f(p) + a
⇒ dp/(b.f(p) + a) = dx
Thus, the equation is separated into two variables p and x which now can be integrated.
Homogeneous Differential Equation
A differential equation of the form dy/dx = f(x,y)/g(x,y) where f and g are functions of the same degree is called Homogeneous Differential Equation. These equations are reduced to a variable separable form by substituting y = vx.
Let’s say dy/dx = f(y/x)/g(y/x) = F (y/x)
Put y = vx
⇒ dy/dx = v + x(dv/dx)
⇒ v + x(dv/dx) = F(v)
⇒ dv/(F(v) – v) = dx/x
The abovementioned form is the variable separable form that can be integrated to get the result. After integration v must be replaced by y/x.
Linear Differential Equation
A differential equation is said to be linear if the dependent variable and its derivative is of first degree. It can be represented in either of the two forms mentioned below:
dy/dx + Ry = S
dx/dy + Px = Q
For the first form, dy/dx + Ry = S, the integrating factor (IF) is e^{∫Rdx}
Solution is ye^{∫Rdx} = ∫Se^{∫Rdx} + C i.e. y.(IF) = ∫S.(IF) + C
For the second form, dx/dy + Px = Q, the integrating factor (IF) is e^{∫Pdx}
Solution is ye^{∫Pdx} = ∫Se^{∫Pdx} + C i.e. y.(IF) = ∫S.(IF) + C
Also, Read
Differential Calculus: FAQs
What is Calculus?
Calculus is the mathematical study of continuous change, analogous to how geometry is the study of shape and algebra is the study of operations and their applications.
Who is the Father of Differential Calculus?
Isaac Barrow is generally credited with the early development of the derivative, but both Isaac Newton and Gottfried Wilhelm Leibniz are considered the fathers of differential calculus.
What are Some Important Topics in Differential Calculus?
Limits, Derivatives, Applications of Derivatives and Differential Equations are some important topics in differential calculus.
What is Derivative?
Derivative measures how a function changes as its input changes, representing the slope of the tangent line to the function’s curve at a given point.
What is Critical Point?
A critical point is where the derivative of a function is zero or undefined, crucial for identifying local extrema and points of inflection.
Define Differential Equation.
A differential equation relates a function’s derivatives to the function itself, describing how it changes over time or in relation to other variables.