Derivatives of Polynomials and Trigonometric Functions

$f\left(x\right)=\sqrt{\sin x}$ then $f^{\text{'}}\left(x\right)$ equals to

If $\tan \left(\frac{x+y}{x-y}\right)=k,$ then $\frac{dy}{dx}$ is equal to

If $y=x\sqrt{x},\frac{dy}{dx}=$

If $y=e^{{\log }_e\left(\sin x\right)}$, then the value of $\frac{dy}{dx}=$

A ball is thrown from the top of a cliff. The path of the ball may be modelled by the equation $y=-0.05x^2+1.5x+82(x\ge 0)$, where $x$ is the horizontal distance from the foot of the cliff and $y$ is the height above the foot of the cliff. Find an expression for $\frac{dy}{dx}$.

Jacek is practising on his skateboard. His journey along the track can be modelled by a curve with equation $f\left(t\right)=10.667-1.667t+0.0417t^2$, where $t$ is the time in seconds and $f\left(t\right)$ is the distance in metres. Find $f^{\text{'}}\left(t\right)$.

A vegetable garden is in the shape of a rectangle. The garden is surrounded by $100$ m of fencing. If the width of the garden is $x$ metres. Find the area of the garden and also find $\frac{dA}{dx}$.

For $f\left(x\right)=x^2-\frac{x}{3}+4$, find $f\text{'}\left(x\right)$.

Jules throws a stone into the air. The height of the stone can be modelled by the equation $f\left(t\right)=1+7.25t-1.875t^2$ where $t$ is the time, in seconds, that has passed since the stone was thrown, and $f\left(t\right)$ is the height of the stone, in metres. Find $f^{\text{'}}\left(t\right)$.

The area, $A$, of a circle of radius $r$ is given by the formula $A=\pi r^2$. Find the rate of change of the area with respect to the radius when $r=2$. Give your answers in terms of $\mathrm{\pi }$.

The area, $A$, of a circle of radius $r$ is given by the formula $A=\pi r^2$. Find $\frac{dA}{dr}$.

Differentiate the following function with respect to $x$,

$y=\frac{7}{x^3}+8x^4-6x^2+2$.

Differentiate the following function with respect to $x$,

$f\left(x\right)=2x\left(x^3-4x-5\right)$.

Differentiate the following function with respect to $x$,

$y=\frac{6}{x^3}+4x-3$.

Differentiate the following function with respect to $x$,

$y=2x^2-\frac{3}{x}$.

Differentiate the following function with respect to $x$,

$f\left(x\right)=5x^4-3x^2+2x-6$.

Differentiate the following function with respect to $x$,

$f\left(x\right)=3x^4+7x-3$.

Differentiate the following function with respect to $x$,

$f\left(x\right)=5x^2-3x$.

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 $\frac{n}{m}$?