Derivative of tan x: Proof by First Principle, Product, Quotient, Chain Rule

The derivative (or differentiation) of tan x is sec² x. In this post, we will learn how to find the derivative of tan x. We will use the limit definition of derivatives as well as the product or quotient rule of derivatives.

What is the Derivative of tan x?

The derivative of tan x is denoted by the symbol d/dx(tan x) or (tan x)$’$ and it is equal to sec² x. Note that tan x can be expressed as (sin x/ cos x). Using this fact together with various trigonometric formulas, we will find the derivative of tan x by the method of

• First principle of derivatives
• Product rule of derivatives
• Quotient rule of derivatives
• Chain rule of derivatives.

Derivative of tan x Formula

Derivative of tan x: The formula of the derivative of tan x is given below

• d/dx(tan x) = sec² x
• (tan x)$’$ = sec² x

Derivative of tan x from limit definition

Derivative of tan x by first principle. We prove that the derivative of tan x is sec² x by limit definition. Let $f(x)=\tan x$. Then the derivative of f(x) from first principle / limit definition is given as follows:

$\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$

Thus we have:

Derivative of tan x by Product Rule

To obtain the derivative of tan x by product rule, let us first recall that rule. For two functions f(x), g(x), the derivative of the product fg is known as the product rule of derivatives which is provided below:

$\dfrac{d}{dx}\left(fg \right)=f \dfrac{dg}{dx} + g \dfrac{df}{dx}$

Proof of the derivative of tan x is sec² x by the product rule of derivatives: we will follow the below steps:

Step 1: At first, we express tan x as the product of two functions as follows

$\tan x=\dfrac{\sin x}{\cos x}=\sin x \cdot \sec x$

$\therefore \dfrac{d}{dx}(\tan x)$ $=\dfrac{d}{dx}(\sin x \cdot \sec x)$

Step 2: Now we use the above product rule of derivatives. So we have

$\dfrac{d}{dx}(\tan x)$ $=\sin x\dfrac{d}{dx}(\sec x)$ $+\sec x \dfrac{d}{dx}(\sin x)$

$=\sin x \cdot \sec x \tan x$ $+\sec x \cdot \cos x$

$=\sin x \cdot \dfrac{1}{\cos x} \cdot \dfrac{\sin x}{\cos x}$ $+\dfrac{1}{\cos x} \cdot \cos x$

$=\dfrac{\sin^2 x}{\cos^2 x}+1$

$=\dfrac{\sin^2 x+\cos^2 x}{\cos^2 x}$

$=\dfrac{1}{\cos^2 x}$ [∵ sin² x+cos² x=1]

$=\sec^2 x$

So the derivative of tan x is sec² x which is obtained by the product rule of derivatives.

Also Read:

Derivative of cotx	The derivative of cotx is -cosec2x.
Derivative of log2x	The derivative of log2x is 1/x.
Derivative of sin2x	The derivative of sin2x is 2sinx cosx.
Derivative of esinx	The derivative of esinx is cosx esinx.

Derivative of tan x by Quotient Rule

Recall, the quotient rule of derivatives: $\dfrac{d}{dx}\left(\dfrac{f}{g} \right)=$ $\dfrac{g\frac{df}{dx}-f\frac{dg}{dx}}{g^2}$, where f and g are two functions of x.

To prove the derivative of tan x is sec² x by the quotient rule of derivatives, we need to follow the below steps.

Step 1: Express tan x as the quotient of two functions. Note that we have

$\tan x=\dfrac{\sin x}{\cos x}$

$\therefore \dfrac{d}{dx}(\tan x)=\dfrac{d}{dx}(\dfrac{\sin x}{\cos x})$

Step 2: Use the above quotient rule of derivatives.

$\dfrac{d}{dx}(\tan x)$ $={\small \dfrac{\cos x \frac{d}{dx}(\sin x)-\sin x\frac{d}{dx}(\cos x)}{\cos^2 x}}$

Step 3: Simplify the above expression.

$=\dfrac{\cos x \cos x-\sin x(-\sin x)}{\cos^2 x}$

$=\dfrac{\cos^2 x+\sin^2 x}{\cos^2 x}$

$=\dfrac{1}{\cos^2 x}$ [∵ sin² x+cos² x=1]

$=\sec^2 x$

So the derivative of tan x by quotient rule is equal to sec² x.

Derivative of tan x by Chain Rule

We know that we can express tan x as 1/cot x. Using this fact we can find the derivative of tan x by the chain rule.

Let z = cot x. So we have dz/dx= -cosec² x

∴ d/dx(tan x) = d/dx(1/cot x) = d/dx(1/z)

⇒ d/dx(tan x) = d/dx(1/z)

= d/dz(1/z) . dz/dx, by the chain rule of derivatives

= d/dz(z^-1) . dz/dx

= (-z^-1-1) . (-cosec² x) by the power rule of derivatives

= 1/z² . cosec² x

= 1/cot² x . cosec² x

= (sin² x)/(cos² x) . 1/sin² x

= 1/ cos²x

= sec² x.

So the derivative of tan x by chain rule is sec² x.

Solved Problems on Derivative of tan x

Question 1: Find the derivative of tan(x²).

Solution:

Let z =x². Thus, dz/dx =2x.

Now the derivative of tan(x²) is

d/dx(tan x²) = d/dz(tan z) · dz/ dx = sec²z · 2x = 2x sec²(x²) as z=x².

Answer: The derivative of tan(x²) is 2x sec²(x²).

Question 2: Find the derivative of tan x with respect to sin x.

Solution:

Let u=tan x and v=sin x. We need to find du/dv.

Now, du/dx = sec²x and dv/dx = cos x ⇒ dx/dv = 1/cos x.

So the derivative of tan x with respect to sin x is

d/d(sin x) (tan x) =du/dv

=du/dx · dx/ dv

= sec²x · 1/cos x = sec²x · sec x = sec³ x.

Answer: the derivative of tan x with respect to sin x is sec³x.

Question 3: Find the derivative of tan x with respect to cot x.

Solution:

Let u=tan x and v=cot x. Here we will find du/dv.

We have du/dx = sec²x and dv/dx = -cosec² x ⇒ dx/dv = -sin²x.

So the derivative of tan x with respect to cot x is

d/d(cot x) (tan x) =du/dv

=du/dx · dx/dv

= sec²x · (- sin² x) = – (sin² x)/(cos² x) = -tan²x.

Answer: the derivative of tan x with respect to cot x is -tan²x.

Question 4: What is the derivative of x tan x?

Solution:

By the chain rule of derivatives, we have

d/dx(x tan x) = x d/dx(tan x) + tan x d/dx(x)

= x sec²x + tan x · 1

= x sec²x + tan x

Answer: The derivative of x tan x is x sec²x + tan x.

FAQs on Derivative of tan x