Derivatives of trigonometric functions
Trigonometric derivatives use radian measure and the chain rule for inner expressions. After differentiating, exact trigonometric values and periodicity help solve any stated condition
\[\frac{d}{dx}\sin(kx)=k\cos(kx),\quad \frac{d}{dx}\cos(kx)=-k\sin(kx)\]
- Use radians Use radians when applying these derivative formulae. Trigonometric equations can be solved in either angle unit if it is used consistently \(\displaystyle 30^\circ=\frac{\pi}{6}\)
- Recall the base derivatives Sine differentiates to cosine, cosine differentiates to negative sine and tangent differentiates to one over cosine squared \(\displaystyle \frac{d}{du}\sin u=\cos u,\quad \frac{d}{du}\cos u=-\sin u,\quad \frac{d}{du}\tan u=\frac{1}{\cos^2u}\)
- Include the inner derivative For a trigonometric function of an inner expression, multiply by the derivative of that inner expression \(\displaystyle \frac{d}{dx}\sin(4x)=4\cos(4x)\)
- Use identities and periods Use exact values, identities and the period of the function to find every required solution in the interval \(\displaystyle \sin^2u+\cos^2u=1,\quad \sin\left(u+2\pi\right)=\sin u\)
Worked example
For the displayed function, find its derivative at the given input
\[h(x)=-2\cos(4x)+3\sin(4x),\quad x=\frac{\pi}{4}\]
- \(\displaystyle u=4x,\quad \frac{du}{dx}=4\)The inner expression in both trigonometric terms is 4x
- \(\displaystyle h'(x)=8\sin(4x)+12\cos(4x)\)Apply the base derivatives and multiply each term by the inner derivative 4
- \(\displaystyle h'\left(\frac{\pi}{4}\right)=8\sin(\pi)+12\cos(\pi)\)Substitute the given input into the inner expression to obtain pi
- \(\displaystyle h'\left(\frac{\pi}{4}\right)=0+12(-1)=-12\)Use the exact sine and cosine values at pi to obtain the derivative value
Common mistakes
Missing the inner coefficient
✕\(\displaystyle \frac{d}{dx}\sin(2x)=\cos(2x)\)✓\(\displaystyle \frac{d}{dx}\sin(2x)=2\cos(2x)\)
The chain rule multiplies by the derivative of the inner expression, so the coefficient of x must be includedSwapping the cosine derivative sign
✕\(\displaystyle \frac{d}{dx}\cos x=\sin x\)✓\(\displaystyle \frac{d}{dx}\cos x=-\sin x\)
The derivative of cosine is negative sine, so the negative sign must be retainedUsing the wrong tangent denominator
✕\(\displaystyle \frac{d}{dx}\tan x=\frac{1}{\sin^2x}\)✓\(\displaystyle \frac{d}{dx}\tan x=\frac{1}{\cos^2x}\)
The tangent derivative is secant squared, which is one over cosine squaredDifficulty levels
introDifferentiate a single sine, cosine or tangent function with a linear input
standardDifferentiate a trigonometric function and evaluate its gradient at a standard angle
Questions about this topic
Why must angles be in radians for these derivative rules?+
The standard calculus derivative rules for sine, cosine and tangent are stated for radian measure
What does the chain rule add to a trigonometric derivative?+
It multiplies the outer derivative by the derivative of the inner expression
How are all solutions found in an interval?+
Use the relevant exact or inverse-trigonometric angle, apply the function's period and include every angle in the stated interval
Is my complete written rule automatically marked?+
No. Written rules and explanations are ungraded
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 6: Applications of differentiation
In this chapter
Differentiate polynomialsChallenge enrichment: rational powers and derivative domainsDifferentiate natural exponential functions and solve simple derivative conditionsDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsDifferentiate a composite function by the chain ruleDifferentiate a product or quotient of two functionsEvaluate and determine function rules6B Differentiation rules: Find the maximal domain of an algebraic functionExpand and factor polynomial expressionsSimplify and combine algebraic fractions6B Differentiation rules: Find the equation of a line from its graph or from its intercept and slopeAnalyse polynomial curves, stationary points and extremaUse a derivative to analyse and recognise the graph of a function6C Tangents, normals and curve sketching: Evaluate a polynomial or recover a coefficient parameter6C Tangents, normals and curve sketching: Move between roots, factors, polynomial rules and equationsRecognise and classify a polynomialConnect a factorised higher-degree rule and its graphAnalyse a cubic graph or determine its ruleWork with sets and interval notation6C Tangents, normals and curve sketching: Find the maximal domain of an algebraic function6C Tangents, normals and curve sketching: Find the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a maximum or minimum problem by differentiationSet up and solve a practical linear equationRearrange literal equations and recognise equivalent line forms6D Optimisation: Find the maximal domain of an algebraic function6D Optimisation: Evaluate a polynomial or recover a coefficient parameter6D Optimisation: Move between roots, factors, polynomial rules and equationsCalculate and interpret average and instantaneous rates from a modelVolume of cones, pyramids and spheresVolume of prisms and cylindersPerimeter and area of standard shapesSimilar figures and scale factorsFind a side of a right-angled triangleAnalyse straight-line motion through displacement, velocity and acceleration6F Kinematics: Evaluate a polynomial or recover a coefficient parameter