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Derivative of Tangent

Complete guide with formula, proof, examples, and graph.

Quick Answer

ddxtan⁡(x)=sec⁡2(x)\frac{d}{dx}\tan(x) = \sec^2(x)

The derivative of \tan(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Express tangent as the quotient of sine and cosine.

tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}

Apply the quotient rule for differentiation.

Applyquotientrule:(uv)′=u′v−uv′v2Apply quotient rule: \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Substitute u = sin(x), v = cos(x), and compute u' and v'.

ddxtan⁡(x)=cos⁡(x)⋅cos⁡(x)−sin⁡(x)⋅(−sin⁡(x))cos⁡2(x)\frac{d}{dx}\tan(x) = \frac{\cos(x)\cdot\cos(x) - \sin(x)\cdot(-\sin(x))}{\cos^2(x)}

Simplify using the Pythagorean identity sin²(x) + cos²(x) = 1.

=cos⁡2(x)+sin⁡2(x)cos⁡2(x)=1cos⁡2(x)=sec⁡2(x)= \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = \frac{1}{\cos^2(x)} = \sec^2(x)

Graph

Visualization of Tangent and its derivative.

f(x) = \tan(x)

f(x)=tan⁡(x)f(x) = \tan(x)

f'(x) = \sec^2(x)

f′(x)=sec⁡2(x)f'(x) = \sec^2(x)
Domain: x≠π2+kπ,k∈Zx \neq \frac{\pi}{2} + k\pi, k \in \mathbb{Z}Range: (−∞,+∞)(-\infty, +\infty)

Worked Examples

Step-by-step solutions using the chain rule and other techniques.

Find: ddxtan⁡(πx)\frac{d}{dx}\tan(\pi x)

Solution: πsec⁡2(πx)\pi \sec^2(\pi x)

1.Chainrulewithu=πx,u′=πChain rule with u = \pi x, u' = \pi
2.ddxtan⁡(πx)=sec⁡2(πx)⋅π\frac{d}{dx}\tan(\pi x) = \sec^2(\pi x) \cdot \pi

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Related Derivatives

sin⁡(x)\sin(x)cos⁡(x)\cos(x)cot⁡(x)\cot(x)sec⁡(x)\sec(x)csc⁡(x)\csc(x)ln⁡(x)\ln(x)log⁡a(x)\log_a(x)x\sqrt{x}arcsin⁡(x)\arcsin(x)

Frequently Asked Questions

What is the derivative of Tangent?

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The derivative of tan⁡(x)\tan(x) is sec⁡2(x)\sec^2(x). This is one of the fundamental derivatives in calculus that you should memorize.

How do you prove the derivative of Tangent?

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The proof uses the limit definition of the derivative. See the Proof section above for the complete step-by-step derivation.

Is the derivative of Tangent always the same?

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Yes, the derivative formula for tan⁡(x)\tan(x) is constant — it does not depend on x. However, when composed with inner functions (e.g., tan⁡(x)\tan(x) of u(x)), the chain rule applies.

Where is the derivative of Tangent undefined?

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The derivative is undefined where the original function is not differentiable. Check the domain section for details.

Why is the derivative of Tangent important?

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This derivative appears frequently in physics (wave motion), engineering (signal processing), economics (oscillating models), and many other fields involving periodic or growth phenomena.