# Total Derivative

• Difficulty Level : Basic
• Last Updated : 19 Jul, 2022

The total derivative of a function f at a point is approximation near the point of function w.r.t. (with respect to) its arguments (variables). Total derivative never approximates the function with a single variable if two or more variables are present in the function. Sometimes, the Total derivative is the same as the partial derivative or ordinary derivative of the function.

### For Composite function:

In general composite, the function is nothing but a function of two or more dependent variables which depend upon any common variable t. Composite function values are obtained from both variables.

If u= f(x,y), where x and y are dependent variables at t, then we can also express u as a function of t. By substituting the value of x, y in f(x,y). Thus, we find the ordinary derivative which is called the total derivative of u

Now, to find without actually substituting the value of x and y in f(x,y).

Similarly, If u = f(x,y,z) where x, y, z are all function of a variable t , then chain rule is:

Question:  Given,   as a function of t.  Verify your result by direct substitution .

Solution: We have,

=

putting values of x and y in above equations

=

Question: Given, f(x,y) =exsiny , x=t3+1 and y=t4+1. Then df/dt at t =1.

Solution: Let f(x,y) =exsiny

= exsiny.(3t2) + cosy .ex .(4t3

As we know , x= t3+1 and y= t4+1

x and y values at t =1, x=2 and y=2

=(2.718)2(0.0349)(12) +(0.9994)(2.718)2(32)

= 238.97

### For Implicit function:

The implicit function is a function whose variables are not completely independent variables. Let a function f(x,y) where x is independent variable but y is x dependent variable.

If f(x, y)= c ( constant )be an implicit function and relation between x and y exists which defines as a differentiable function of x

Here, f(x,y) =constant

For implicit function let us consider, x an independent variable, and y is a function of x

f(x,y) = c           ……..eq (1)

by definition of total differential coefficient.

Question: If u = xlogxy where x3 +y3+3xy=1, find du/dx.

Solution: We have x3 +y3+3xy=1    ……….(1)

=

from eq……….(1)

after putting value in eq (2)

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