Question

Differentiate the following function with respect to x.
${{x}^{n}}\tan x$ .
(a) ${{x}^{n-1}}\left( n\tan x+x\sec x \right)$
(b) ${{x}^{n-1}}\left( n\tan x+x{{\sec }^{2}}x \right)$
(c) ${{x}^{n-1}}\left( n\tan x+\sec x \right)$
(d) ${{x}^{n-1}}\left( n\tan x+x{{\sec }^{-2}}x \right)$

Answer 1

Hint: In order to crack this problem, we need to know the product rule of differentiation. It is shown as follows, $\dfrac{df\left( x \right)}{dx}=h\left( x \right)\dfrac{dg\left( x \right)}{dx}+g\left( x \right)\dfrac{dh\left( x \right)}{dx}$ . Also, we need to know the individual differentiation formulas for each sun functions like $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$ and $\dfrac{d\tan x}{dx}={{\sec }^{2}}x$ .

Complete step-by-step answer:
We aim to find the differentiation of this function with respect to x.
Let the function be named as $f\left( x \right)={{x}^{n}}\tan x$ .
The function $f\left( x \right)$ contains two sub-functions.
Let $g\left( x \right)={{x}^{n}}$ and $h\left( x \right)=\tan x$ be the two sub-functions.
Therefore, $f\left( x \right)=g\left( x \right)\times h\left( x \right).....................\left( i \right)$ .
To find the differentiation we need to use the product rule of differentiation as the function is the products of two spate functions.
The product rule states as follows,
$\dfrac{df\left( x \right)}{dx}=h\left( x \right)\dfrac{dg\left( x \right)}{dx}+g\left( x \right)\dfrac{dh\left( x \right)}{dx}................\left( ii \right)$
By substituting the values of $f\left( x \right),g\left( x \right),h\left( x \right)$ we get,
$\dfrac{df\left( x \right)}{dx}=\tan x\dfrac{d{{x}^{n}}}{dx}+{{x}^{n}}\dfrac{d\tan x}{dx}$
The differentiation ${{x}^{n}}$ is $\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}..............(iii)$
And, the differentiation of $\tan x$ is $\dfrac{d\tan x}{dx}={{\sec }^{2}}x................(iv)$
Substituting the values from equation (iii) and (iv) we get,
$\dfrac{df\left( x \right)}{dx}=\tan x\left( n{{x}^{n-1}} \right)+{{x}^{n}}\left( {{\sec }^{2}}x \right)$
Simplifying it further we get,
$\dfrac{df\left( x \right)}{dx}={{x}^{n-1}}\left( n\tan x+x{{\sec }^{2}}x \right)$ .
Hence the option that matches is option (b).


Note: One of the most important points in this problem is to remember the differentiation formulas. The common point of mistake is where we simplify the equation and we take ${{x}^{n-1}}$ coming out of brackets. Therefore, when ${{x}^{n-1}}$ is common, there is $x$ with ${{\sec }^{2}}x$ . Also, option (c) is exactly the mistake that is commonly made. As the options are very close we need to be extra careful with this sum.