The main purpose of this question is to find the differential of each given function.

A function is a fundamental mathematical concept that describes a relationship between a set of inputs and a set of possible outputs, with each input corresponding to one output. The input is an independent variable and the output is referred to as a dependent variable.

Differential calculus and integral calculus are the fundamental classifications of calculus. Differential calculus deals with infinitely small changes in some varying quantity. Let $y=f(x)$ be a function with a dependent variable $y$ and an independent variable $x$. Let $dy$ and $dx$ be the differentials. The differential forms the main part of the change in a function $y = f(x)$ as the independent variable changes. The relation between $dx$ and $dy$ is given by $dy=f'(x)dx$.

More generally, differential calculus is used to investigate the instantaneous rate of change, for instance, velocity, to estimate the value of a small variation in a quantity, and to determine whether a function in a graph is increasing or decreasing.

## Expert Answer

(a) The given function is:

$y=\tan(\sqrt{7t})$

or $y=\tan(7t)^{1/2}$

Here, $y$ is dependent and $t$ is an independent variable.

Taking differential of both sides using the chain rule as:

$dy=\sec^2(7t)^{1/2}\cdot\dfrac{1}{2}(7t)^{-1/2}(7)\,dt$

Or $dy=\dfrac{7\sec^2(\sqrt{7t})}{2\sqrt{7t}}\,dt$

(b) The given function is:

$y=\dfrac{3-v^2}{3+v^2}$

Here, $y$ is dependent and $v$ is an independent variable.

Taking differential of both sides using the quotient rule as:

$dy=\dfrac{(3+v^2)\cdot(-2v)-(3-v^2)(2v)}{(3+v^2)^2}\,dv$

$dy=\dfrac{-6v-v^3-6v+2v^3}{(3+v^2)^2}\,dv$

$dy=\dfrac{-12v}{(3+v^2)^2}\,dv$

## Examples

Find the differential of the following functions:

(a) $f(y)=y^2-\sec(y)$

Using the power rule on first term and the chain rule on second term as:

$df(y)=[2y-\sec(y)\tan(y)]\,dy$

(b) $y=x^4-9x^2+12x$

Using power rule on all the terms as:

$dy=(4x^3-18x+12)\,dx$

(c) $h(x)=(x-2)(x-x^3)$

Rewrite the function as:

$h(x)=x^2-x^4-2x+2x^3$

$h(x)= -x^4+2x^3+x^2-2x$

Now use the power rule on all the terms as:

$dh(x)=( -4x^3+6x^2+2x-2)\,dx$

(d) $x=\dfrac{3}{\sqrt{t^3}}+\dfrac{1}{4t^4}-\dfrac{1}{t^{11}}$

Rewrite the given function as:

$x=3t^{-3/2}+\dfrac{1}{4}t^{-4}-t^{-11}$

Now use power rule on all the terms as:

$dx=\left(-\dfrac{9}{2}t^{-1/2}-t^{-3}+11t^{-10}\right)\,dt$

$dx=\left(-\dfrac{9}{2\sqrt{t}}-\dfrac{1}{t^3}+\dfrac{11}{t^{10}}\right)\,dt$

(e) $y=\ln(\sin (2x))$

Using the chain rule as:

$dy=\dfrac{1}{\sin(2x)}\cdot\cos(2x)\cdot 2\,dx$

$dy=\dfrac{2\cos(2x)}{\sin(2x)}\,dx$

Or $dy=2\cot(2x)\,dx$

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