Class 12 RD Sharma Solutions – Chapter 11 Differentiation – Exercise 11.1

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 x²e^x

Solution:

We have,

Let,

f(x)=x²e^x

f(x+h)=(x+h)²e^(x+h)

Since, 

=x²e^x+2xe^x+0

=e^x(x²+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