Derivative of tan square x | tan^2x derivative

The derivative of tan square x is equal to 2tanx sec²x. Here we find the derivative of tan²x by the chain and product rule. The derivative formula of tan square x is given below:

${\displaystyle \frac{d}{dx}}({\tan }^{2}x)=2\tan x{\sec }^{2}x$.

What is the derivative of tan square x?

Let us find the derivative of tan²x by the chain rule. To do so, let us put

z=tanx.

So ${\displaystyle \frac{dz}{dx}}={\sec }^{2}x$.

Now, by the chain rule, the derivative of tan square x will be equal to

${\displaystyle \frac{d}{dx}}({\tan }^{2}x)$ = ${\displaystyle \frac{d}{dx}}(z^2)\times {\displaystyle \frac{dz}{dx}}$

= 2z × sec²x by the power rule of integration: d/dx(xⁿ) = nx^n-1.

= 2tanx sec²x as z=tanx.

So the derivative of tan square x is equal to 2tanx sec²x, and this is proved by the chain rule of derivatives.

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Derivative of tan²x by Product Rule

Note that tan²x is a product of two times of tanx. So the derivative of tan²x by the product rule is equal to

${\displaystyle \frac{d}{dx}}({\tan }^{2}x)$ = ${\displaystyle \frac{d}{dx}}(\tan x\cdot \tan x)$

= tan x ${\displaystyle \frac{d}{dx}}(\tan x)$ + tan x ${\displaystyle \frac{d}{dx}}(\tan x)$

= tanx sec²x + tanx sec²x as the derivative of tanx is sec²x.

= 2tanx sec²x.

So the derivative of tan²x is tanx sec²x, and this is obtained by the product rule of derivatives.

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