**Question 1.** Differentiate the following functions from first principles e^{-x}

**Solution:**

We have,

Let,

f(x)=e

^{-x}f(x+h)=e

^{-(x+h)}=-e

^{-x}

**Question 2. **Differentiate the following functions from first principles e^{3x}

**Solution:**

We have,

Let,

f(x)=e

^{3x}f(x+h)=e

^{3(x+h)}=3e

^{3x}

**Question 3.** Differentiate the following functions from first principles e^{ax+b}

**Solution:**

We have,

Let,

f(x)=e

^{ax+b}f(x+h)=e

^{a(x+h)+b}=ae

^{ax+b}

**Question 4.** Differentiate the following functions from first principles e^{cosx}

**Solution:**

We have,

Let,

f(x)=e

^{cosx}f(x+h)=e

^{cos(x+h)}=e

^{cosx}(-sinx)=-sinx.e

^{cosx}

**Question 5.** Differentiate the following functions from first principles e^{√2x}

**Solution:**

We have,

Let,

f(x)=e

^{√2x}f(x+h)=e

^{√2(x+h)}(After rationalising the numerator)

**Question 6.** Differentiate the following functions from first principles log(cosx)

**Solution:**

We have,

Let,

f(x)=log(cosx)

f(x+h)=log(cos(x+h))

Since,

=-(2sinx)/(2cosx)

=-tanx

**Question 7.** Differentiate the following functions from first principles e^{√cotx}

**Solution:**

We have,

Let,

f(x)=e

^{√cotx}f(x+h)=e

^{√cot(x+h)}since,

(After rationalising the numerator)

Since,

**Question 8.** Differentiate the following functions from first principles x^{2}e^{x}

**Solution:**

We have,

Let,

f(x)=x

^{2}e^{x}f(x+h)=(x+h)

^{2}e^{(x+h)}Since,

=x

^{2}e^{x}+2xe^{x}+0=e

^{x}(x^{2}+2x)

**Question 9.** Differentiate the following functions from first principles log(cosecx)

**Solution:**

We have,

Let,

f(x)=log(cosecx)

f(x+h)=log(cosec(x+h))

=-cotx

**Question 10.** Differentiate the following functions from first principles sin^{-1}(2x+3)

**Solution:**

We have,

Let,

f(x)=sin

^{-1}(2x+3)f(x+h)=sin

^{-1}[2(x+h)+3]f(x+h)=sin

^{-1}(2x+2h+3)Where

(After rationalising the numerator)

Solving above equation