Question

What is the derivative of ${\tan ^2}x\sec x$?

Answer 1

To solve this question first we assume a function equal to the given relation then we find the derivative with respect to $x$. To find the derivative of the function we have to use the multiplication rule of the derivative. Then we use the identity of trigonometry and convert it into the tan trigonometry function.

Complete step by step solution:
Let the given function is assumed as-
$y = {\tan ^2}x\sec x$
Now we have to find the derivative of this function with respect to $x$.
Differentiating both side
$\dfrac{{d\left( y \right)}}{{dx}} = \dfrac{{d\left( {{{\tan }^2}x\sec x} \right)}}{{dx}}$
Now using the multiplication rule of derivative $\dfrac{{d\left( {f\left( x \right)g\left( x \right)} \right)}}{{dx}} = f\left( x \right)\dfrac{{d\left( {g\left( x \right)} \right)}}{{dx}} + g\left( x \right)\dfrac{{d\left( {f\left( x \right)} \right)}}{{dx}}$
$\dfrac{{d\left( y \right)}}{{dx}} = {\tan ^2}x\dfrac{{d\left( {\sec x} \right)}}{{dx}} + \sec x\dfrac{{d\left( {{{\tan }^2}x} \right)}}{{dx}}$
We know the derivative of $\sec x$ is $\tan x\sec x$ and derivative of ${\tan ^2}x$ is $2\tan x{\sec ^2}x$
On putting these values in the equation.
$\dfrac{{d\left( y \right)}}{{dx}} = {\tan ^2}x\tan x\sec x + \sec x2\tan x{\sec ^2}x$
Now on simplifying this equation.
$\dfrac{{d\left( y \right)}}{{dx}} = {\tan ^3}x\sec x + 2\tan x{\sec ^3}x$
On taking the common part common from the equation
$\dfrac{{d\left( y \right)}}{{dx}} = \tan x\sec x\left( {{{\tan }^2}x + 2{{\sec }^2}x} \right)$
Now converting ${\sec ^2}x$ into ${\tan ^2}x$ using the trigonometry identity.
$1 + {\tan ^2}x = {\sec ^2}x$on putting this value in the equation.
$\dfrac{{d\left( y \right)}}{{dx}} = \tan x\sec x\left( {{{\tan }^2}x + 2\left( {1 + {{\tan }^2}x} \right)} \right)$
On further solving
$\dfrac{{d\left( y \right)}}{{dx}} = \tan x\sec x\left( {{{\tan }^2}x + 2 + 2{{\tan }^2}x} \right)$
On adding the like terms
$\dfrac{{d\left( y \right)}}{{dx}} = \tan x\sec x\left( {3{{\tan }^2}x + 2} \right)$
The derivative of the function ${\tan ^2}x\sec x$ is:
$\dfrac{{d\left( y \right)}}{{dx}} = \tan x\sec x\left( {3{{\tan }^2}x + 2} \right)$

Note:
To solve these types of questions students must know the multiplication and division rule of derivatives. And the identities of trigonometry. Students often make mistakes in the multiplication rule. The final answer must be changed according to the options if they are given in the question. These changes are made with the help of identities and formulas of trigonometry.
Differentiation indicates the rate of change of a quantity with respect to another variable. Integration indicates the sum of small elements. Differentiation and integration both are opposite to each other. Differentiation of integration is the same function.