Question

Differentiate the following functions with respect to x : (i) x^(3)+2x^(2)+1 " " (ii) e^((3x+2)) " " (iii) cos^(2) x

Answer 1

Answer:

Refer the attached picture.

Answer 2

Answer:

In this type of complex functions , try to apply chain rule and then find the derivative.

$\red{ \sf{1st \: question}}$

$y = {x}^{3} + 2 {x}^{2} + 1$

$= > \dfrac{dy}{dx} = 3 {x}^{2} + 4x + 0$

$= > \dfrac{dy}{dx} = 3 {x}^{2} + 4x$

$\red{ \sf{2nd\: question}}$

$y = {e}^{3x + 2}$

$= > \dfrac{dy}{dx} = \dfrac{d \: {e}^{(3x + 2)} }{d(3x + 2)} \times \dfrac{d(3x + 2)}{dx}$

$= > \dfrac{dy}{dx} = 3 {e}^{(3x + 2)}$

$\red{ \sf{3rd\: question}}$

$y = {cos}^{2} (x)$

$= > \dfrac{dy}{dx} = \dfrac{d \: {cos}^{2}(x) }{d \: cos(x)} \times \dfrac{d \: cos(x)}{dx}$

$= > \dfrac{dy}{dx} = 2 \cos(x) \times \{ - \sin(x) \}$

$= > \dfrac{dy}{dx} = - \{ 2 \cos(x) \times \sin(x) \}$

$= > \dfrac{dy}{dx} = - \sin(2x)$