Question

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

Answer 1

Hint: If we need to find the derivative of the function ${{f}^{n}}\left( x \right),$ we will first differentiate this function for the exponent and then we will differentiate the function. Mathematically, we write $\dfrac{d}{dx}{{f}^{n}}\left( x \right)=n{{f}^{n-1}}\left( x \right){f}'\left( x \right).$ The derivative of $\sec x$ is $\sec x\tan x.$

Complete step by step solution:
Let us consider the given problem.
We are asked to find the derivative of the given function.
The given function we need to differentiate is $y={{\sec }^{2}}x.$
Before start differentiating the given function, we should know the rule of differentiation given by $\dfrac{d}{dx}{{f}^{n}}\left( x \right)=n{{f}^{n-1}}\left( x \right){f}'\left( x \right).$
So, we will differentiate the given function considering that the given function is of the form ${{x}^{n}}.$ Then, we will differentiate the function regardless of the exponent.
So, we can write $\dfrac{dy}{dx}=\dfrac{d}{dx}{{\sec }^{2}}x.$
Also, we should know that the derivative of $\sec x$ is $\sec x\tan x.$
Let us suppose that $f\left( x \right)=\sec x.$ Then, we will get ${{f}^{2}}\left( x \right)={{\sec }^{2}}x.$
We will get, $2{{f}^{2-1}}\left( x \right)=2f\left( x \right)=2\sec x.$
Similarly, we will get ${f}'\left( x \right)=\sec x\tan x.$
Now the first part of the derivative will be $2\sec x.$
The second part of the derivative will be $\sec x\tan x.$
That is, we will get $\dfrac{dy}{dx}=\dfrac{d}{dx}{{\sec }^{2}}x=2\sec x\sec x\tan x.$
That is, $\dfrac{dy}{dx}=\dfrac{d}{dx}{{\sec }^{2}}x=2\sec x\sec x\tan x=2{{\sec }^{2}}x\tan x.$
Since $y={{\sec }^{2}}x,$ we will substitute this in the above equation.
So, we will get the derivative of the given function as $\dfrac{dy}{dx}=2{{\sec }^{2}}x\tan x=2y\tan x.$
Hence the derivative of the given function is obtained as $\dfrac{dy}{dx}=2y\tan x.$

Note: Let us recall the derivatives of the basic trigonometric functions. The derivative of $\sin x$ is $\cos x.$ The derivative of $\cos x$ is $-\sin x.$ The derivative of $\tan x$ is ${{\sec }^{2}}x.$ The derivative of $\sec x$ is $\sec x\tan x.$ The derivative of $\cot x$ is $-\cos e{{c}^{2}}x.$ The derivative of $\cos ecx$ is $-\cos ecx\cot x.$ When we differentiate a function, we should first consider the exponent, if any.