Derivatives of Trigonometric Functions
The expression contains a trigonometric function.
Apply its standard rule, then differentiate the angle.
Check signs and reconstruct with sine and cosine if unsure.
Calculus is the language of change; trigonometry is the language of cycles and oscillation. Where the two intersect, we arrive at the derivatives of trigonometric functions — a set of results that lets us measure the very rhythm of change itself.
The Bedrock: First Principles and Foundational Limits
One critical prerequisite before deriving anything: all angles must be measured in radians. The clean derivative formulas below only hold in radians — in degrees, awkward conversion constants appear everywhere.
The cornerstone limit: \(\lim_{\theta\to 0} \frac{\sin\theta}{\theta} = 1\)
The route to the derivative of sine runs through this limit, proved with a beautiful piece of geometric reasoning: the Squeeze Theorem.
Squeeze Theorem proof sketch
- Geometric construction: on a unit circle, take a small angle \(\theta\) and compare three regions — the triangle inside the arc, the circular sector itself, and the larger triangle formed by the tangent line.
- Area comparison: the inner triangle has area \(\frac{1}{2}\sin\theta\), the sector has area \(\frac{1}{2}\theta\), and the outer triangle has area \(\frac{1}{2}\tan\theta\).
- Establish the inequality: \[\frac{1}{2}\sin\theta < \frac{1}{2}\theta < \frac{1}{2}\tan\theta\]
- Algebraic manipulation: dividing through by \(\frac{1}{2}\sin\theta\) and taking reciprocals gives \[\cos\theta < \frac{\sin\theta}{\theta} < 1\]
- Apply the Squeeze Theorem: as \(\theta \to 0\), \(\cos\theta \to 1\), so \(\frac{\sin\theta}{\theta}\) is squeezed between two quantities that both approach 1 — and must equal 1 in the limit.
The corollary limit: \(\lim_{\theta\to 0} \frac{\cos\theta - 1}{\theta} = 0\)
Multiplying by the conjugate turns this into a form the cornerstone limit can handle:
\[\lim_{\theta\to 0} \frac{\cos\theta - 1}{\theta} = \lim_{\theta\to 0} \frac{\cos^2\theta - 1}{\theta(\cos\theta + 1)} = \lim_{\theta\to 0}\left(-\frac{\sin\theta}{\theta}\cdot\frac{\sin\theta}{\cos\theta+1}\right) = -1 \cdot \frac{0}{2} = 0\]Deriving the derivatives of sine and cosine
With both limits established, the limit definition of the derivative and the angle-addition formulas deliver the two foundational results:
\[\frac{d}{dx}(\sin x) = \lim_{h\to 0} \frac{\sin(x+h) - \sin x}{h} = \cos x\] \[\frac{d}{dx}(\cos x) = \lim_{h\to 0} \frac{\cos(x+h) - \cos x}{h} = -\sin x\]Building the Family: The Remaining Four
Once the derivatives of \(\sin x\) and \(\cos x\) are established, all other trigonometric derivatives follow systematically from the quotient rule (see the product and quotient rules lesson).
| Function \(f(x)\) | Derivative \(f'(x)\) | Chain rule form \(\frac{d}{dx}[f(u)]\) |
|---|---|---|
| \(\sin x\) | \(\cos x\) | \(\cos(u)\cdot u'\) |
| \(\cos x\) | \(-\sin x\) | \(-\sin(u)\cdot u'\) |
| \(\tan x\) | \(\sec^2 x\) | \(\sec^2(u)\cdot u'\) |
| \(\cot x\) | \(-\csc^2 x\) | \(-\csc^2(u)\cdot u'\) |
| \(\sec x\) | \(\sec x\tan x\) | \(\sec(u)\tan(u)\cdot u'\) |
| \(\csc x\) | \(-\csc x\cot x\) | \(-\csc(u)\cot(u)\cdot u'\) |
The Other Side of the Angle: Inverse Trig Derivatives
Inverse trigonometric functions ask, "What angle produced this value?" Their derivatives are found using implicit differentiation (see the implicit differentiation lesson).
| Function \(f(x)\) | Derivative \(f'(x)\) | Domain of \(f(x)\) |
|---|---|---|
| \(\arcsin x\) | \(\frac{1}{\sqrt{1 - x^2}}\) | \([-1, 1]\) |
| \(\arccos x\) | \(-\frac{1}{\sqrt{1 - x^2}}\) | \([-1, 1]\) |
| \(\arctan x\) | \(\frac{1}{1 + x^2}\) | \((-\infty, \infty)\) |
| \(\operatorname{arccot} x\) | \(-\frac{1}{1 + x^2}\) | \((-\infty, \infty)\) |
| \(\operatorname{arcsec} x\) | \(\frac{1}{|x|\sqrt{x^2 - 1}}\) | \((-\infty, -1] \cup [1, \infty)\) |
| \(\operatorname{arccsc} x\) | \(-\frac{1}{|x|\sqrt{x^2 - 1}}\) | \((-\infty, -1] \cup [1, \infty)\) |
From foundational geometric proofs to the oscillating systems of the physical world, the trigonometric derivatives form a tightly interconnected family — each result flowing from two limits, one squeeze, and the rules covered in the guides above. That interconnectedness is a testament to the unifying beauty and utility of mathematics.
Try it: differentiate tan(x) and see the quotient rule derive sec²x →Domains and inverse-function branches
Sine and cosine are defined and differentiable for all real inputs. Tangent and secant exclude x = π/2 + kπ; cotangent and cosecant exclude x = kπ, for every integer k. Apply the same restrictions to the inner argument for composite functions.
The inverse-function table lists function domains. Finite derivatives of arcsin and arccos require |x| < 1; those of arcsec and arccsc require |x| > 1. Endpoint function values do not imply finite endpoint derivatives. Arctan is differentiable for every real input.
The displayed arccot formula uses its decreasing branch with range (0, π). The arcsec formula uses range [0, π] excluding π/2, and arccsc uses [−π/2, π/2] excluding zero. Alternative branch conventions must be handled consistently.
The geometric squeeze argument starts with 0 < θ < π/2. Since sin(θ)/θ is even, the negative-angle limit agrees, establishing the two-sided limit. In degree coordinates t, write sin(πt/180) first; its derivative includes the factor π/180.
Compare the secant derivation and arcsine derivation to distinguish reciprocal and inverse functions.
Check your understanding
- Differentiate sin(4x).
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4cos(4x).
- Differentiate x tan(x).
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tan(x)+x sec²(x).
- Differentiate csc(2x).
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−2csc(2x)cot(2x).
Reference standard: Rule statements and notation follow OpenStax Calculus Volume 1, Chapter 3. DerivCalc's explanations and examples are independently written.