**Question 1. Find the derivative of f(x) = 3x at x = 2**

**Solution:**

Given: f(x)=3x

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)=3x at x=2 is given as:

⇒

⇒

⇒

⇒

Hence, derivative of f(x)=3x at x=2 is 3

**Question 2. Find the derivative of f(x) = x**^{2}– 2 at x = 10

^{2}– 2 at x = 10

**Solution:**

Given: f(x)= x

^{2}-2By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)=x

^{2}-2 at x=10 is given as:⇒

⇒

⇒

⇒

⇒

Hence, derivative of f(x)=x

^{2}-2 at x=10 is 20

**Question 3. Find the derivative of f(x) = 99x at x = 100**

**Solution:**

Given: f(x)= 99x

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)=99x at x=100 is given as:

⇒

⇒

⇒

Hence, derivative of f(x)=99x at x=100 is 99

**Question 4. Find the derivative of f(x) = x at x = 1**

**Solution:**

Given: f(x)=x

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)=x at x=1 is given as:

⇒

⇒

⇒

Hence, derivative of f(x)=x at x=1 is 1

**Question 5. Find the derivative of f(x) = **** at x = 0**

**Solution:**

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at x=0 is given as:

⇒

⇒

⇒

∵ we can not find the limit of the above function f(x)= by direct substitution as it gives 0/0 form (indeterminate form)

So we will simplify it to find the limit.

As we know that

∴

Divide the numerator and denominator by 2 to get the form for applying sandwich theorem and multiplying h in numerator and denominator to get the required form.

⇒

⇒

Using the formula:

∴

Hence, derivative of f(x)= at x=0 is 0

**Question 6. Find the derivative of f(x) = ** **at x = 0**

**Solution:**

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at x=0 is given as:

⇒

⇒

⇒

∴ Use the formula: {sandwich theorem}

⇒

Hence, derivative of f(x)= at x=0 is 1

**Question 7(i). Find the derivatives of the following functions at the indicated points :** ** at**

**Solution:**

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at is given as:

⇒

⇒

⇒ f'(\pi/2)= {∵

∵ we can not find the limit of the above function by direct substitution as it gives 0/0 form (indeterminate form)

So we will simplify it to find the limit.

As we know that

∴

Divide the numerator and denominator by 2 to get the form (sin x)/x for applying sandwich theorem and multiplying h in numerator and denominator to get the required form.

⇒

⇒

Using the formula:

∴

Hence, derivative of f(x)= at is 0

**Question 7(ii). Find the derivatives of the following functions at the indicated points : x at x=1**

**Solution:**

Given: f(x)=x

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)=x at x=1 is given as:

⇒

⇒

⇒

Hence, derivative of f(x)=x at x=1 is 1

**Question 7(iii). Find the derivatives of the following functions at the indicated points : **2\cos x **at**

**Solution:**

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at is given as:

⇒

⇒ f'(\pi/2)= \lim_{h \to 0} \frac {-2\sin(h)} h {∵ }

∵ we can not find the limit of the above function by direct substitution as it gives 0/0 form (indeterminate form)

∴

Using the formula:

∴

Hence, derivative of f(x)=

**Question 7(iv). Find the derivatives of the following functions at the indicated points :** **at**

**Solution:**

Given: f(x)=

By using the derivative formula,

{where h is a small positive number}

Derivative of f(x)= at is given as:

⇒

⇒ {∵}

⇒

⇒

∵ we can not find the limit of the above function by direct substitution as it gives 0/0 form (indeterminate form)

Using the sandwich theorem and multiplying 2 in numerator and denominator to apply the formula.

Using the formula:

∴

Hence, derivative of f(x)=