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Derivative of Secant

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

Quick Answer

ddxsec⁡(x)=sec⁡(x)tan⁡(x)\frac{d}{dx}\sec(x) = \sec(x)\tan(x)

The derivative of \sec(x) is:

Proof / Derivation

Step-by-step derivation of the derivative formula.

Rewrite secant as a power of cosine.

sec⁡(x)=1cos⁡(x)=[cos⁡(x)]−1\sec(x) = \frac{1}{\cos(x)} = [\cos(x)]^{-1}

Apply the chain rule with the power rule.

Applychainrule:ddx[u]−1=−u−2⋅u′Apply chain rule: \frac{d}{dx}[u]^{-1} = -u^{-2} \cdot u'

Differentiate the outer function and multiply by the derivative of cos(x).

ddxsec⁡(x)=−[cos⁡(x)]−2⋅(−sin⁡(x))\frac{d}{dx}\sec(x) = -[\cos(x)]^{-2} \cdot (-\sin(x))

Simplify by factoring into sec(x) and tan(x).

=sin⁡(x)cos⁡2(x)=1cos⁡(x)⋅sin⁡(x)cos⁡(x)=sec⁡(x)tan⁡(x)= \frac{\sin(x)}{\cos^2(x)} = \frac{1}{\cos(x)} \cdot \frac{\sin(x)}{\cos(x)} = \sec(x)\tan(x)

Graph

Visualization of Secant and its derivative.

f(x) = \sec(x)

f(x)=sec⁡(x)f(x) = \sec(x)

f'(x) = \sec(x)\tan(x)

f′(x)=sec⁡(x)tan⁡(x)f'(x) = \sec(x)\tan(x)
Domain: x≠π2+kπ,k∈Zx \neq \frac{\pi}{2} + k\pi, k \in \mathbb{Z}Range: (−∞,−1]∪[1,+∞)(-\infty, -1] \cup [1, +\infty)

Worked Examples

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

Find: ddxsec⁡(3x+1)\frac{d}{dx}\sec(3x+1)

Solution: 3sec⁡(3x+1)tan⁡(3x+1)3\sec(3x+1)\tan(3x+1)

1.u=3x+1,u′=3u = 3x+1, u' = 3
2.=3sec⁡(3x+1)tan⁡(3x+1)= 3\sec(3x+1)\tan(3x+1)

Calculate Any Derivative

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Related Derivatives

sin⁡(x)\sin(x)cos⁡(x)\cos(x)tan⁡(x)\tan(x)cot⁡(x)\cot(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 Secant?

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

How do you prove the derivative of Secant?

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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 Secant always the same?

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

Where is the derivative of Secant 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 Secant 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.