Differentiation of x to the Power n

Differentiate the following function with respect to $x$,

$y={\left(x^2+2\right)}^2$.

A particle starts rotating from rest according to the formula $\theta =\frac{t^2}{64}-\frac{t}{8}$ where $\theta$ is the angle in radian and t in seconds. Find the angular velocity and angular acceleration at the end of 4 s.

The area of dot of ink in $c{m}^2$ is growing such that after 5s, $A = 3t^2 + \left(\frac{t}{5}\right) + 7$. Calculate the rate of increases of the area after $5 s$.

A particle starts rotating from rest according to the formula $\theta =\frac{3t^2}{20}-\frac{t^2}{3}$. Find the angular velocity and the acceleration at the end of 5s

 A stone is dropped into a quiet pond and waves move in circles at a speed of$4 cm s^{-1}$. At the instant when the radius of the circular wave is $10 cm$, how fast is the enclosed area increasing?

The side of a square sheet of metal is increasing at $3$ cm per minute. At what rate the area is increasing when the side is $10$ cm long?

A car is running on a straight road. The distance  travelled and time taken are connected by  the formula $s=t^2-2t$ where $t$ is measured in hours and $s$ in $km$. When the odometer reading is $15$, what is the speedometer reading?  [Odometer measures $s$ and the speedometer measures velocity]

At time t, the displacement of a particle moving in a straight line is given by $x=-4t^2+ 2t$. Find the velocity and accleration a when $t = \left(\frac{1}{2}\right) s$.

Differentiate wrt x using product rule

(i) $\left(5x+2\right)\left(4-3x\right)$

Differentiate the following with respect to x :

(vi) $\tan x +2 \sin x+3\cos x-\frac{1}{2}\log x-e^x$

Differentiate the following  with respect to x :

(i) $x^2+3x+\frac{3}{x}$

Differentiate the following  with respect to x :

(i) $4x^3-10-6x^2$

Differentiate w.r.t. x

(i) $x^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Differentiate w.r.t. x

(i) $x^{-4}$

Differentiate w.r.t. x

(i) $16x^8$

Differentiate the following w.r.t. x

(1) $e^{-5}$

Differentiate the following w.r.t. x

(1) $\mathrm{\pi e}$

Differentiate the following w.r.t. x

(1) ${\mathrm{\pi }}^2$

Differentiate the following w.r.t. x

(1) 2020

If the derivative of $f\left(x\right)=x^{-3}(5+3x)$ with respect to $x$ is $f^{\text{'}}\left(x\right)=m{x}^{-4}+n{x}^{-3}$. Finf the value of $\frac{n}{m}$?