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Differentiate sine, cosine and tangent functions and solve simple derivative conditions

Start a set of 8
Step 1 of 5

For the function shown, which first step is correct?

\[f(x)=4\sin(3x)\]
Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

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

\[\frac{d}{dx}\sin(kx)=k\cos(kx),\quad \frac{d}{dx}\cos(kx)=-k\sin(kx)\]
  1. Use radians Use radians when applying these derivative formulae. Trigonometric equations can be solved in either angle unit if it is used consistently \(\displaystyle 30^\circ=\frac{\pi}{6}\)
  2. Recall the base derivatives Sine differentiates to cosine, cosine differentiates to negative sine and tangent differentiates to one over cosine squared \(\displaystyle \frac{d}{du}\sin u=\cos u,\quad \frac{d}{du}\cos u=-\sin u,\quad \frac{d}{du}\tan u=\frac{1}{\cos^2u}\)
  3. Include the inner derivative For a trigonometric function of an inner expression, multiply by the derivative of that inner expression \(\displaystyle \frac{d}{dx}\sin(4x)=4\cos(4x)\)
  4. 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

Find the derivative of the function shown at the given input

\[h(x)=-2\cos(4x)+3\sin(4x),\quad x=\frac{\pi}{4}\]
  1. \(\displaystyle u=4x,\quad \frac{du}{dx}=4\)Both trigonometric terms have the same inner expression
  2. \(\displaystyle h'(x)=8\sin(4x)+12\cos(4x)\)Apply the base derivatives and multiply each term by the inner derivative
  3. \(\displaystyle h'\left(\frac{\pi}{4}\right)=8\sin(\pi)+12\cos(\pi)\)Substitute the given input into the derivative
  4. \(\displaystyle h'\left(\frac{\pi}{4}\right)=0+12(-1)=-12\)Use the exact sine and cosine values 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 included
Swapping 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 retained
Using 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 squared

Difficulty 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
challengeSimplify a trigonometric derivative using an identity or use it in a line condition
examSolve a derivative equation over an interval or use the derivative in a tangent or normal line problem

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

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 5: Differentiation

In this chapter

Evaluate limits and test continuityGradient of a tangent from first principlesDifferentiate power functions and evaluate a derivativeExpand and factor polynomial expressionsPascal rows and binomial coefficient structure5A Limits, first principles and the power rule: Simplify and combine algebraic fractionsEvaluate a polynomial or recover a coefficient parameterSubstitute values into a formula5A Limits, first principles and the power rule: Solve a linear equationTest or construct a smooth piecewise joinRead and construct a gradient function from a function graphAnalyse a cubic graph or determine its ruleEvaluate and determine function rulesEquation of a tangent or normal lineRecover a polynomial rule from derivative and tangent conditionsFind the gradient of a line through two pointsFind the equation of a line from its graph or from its intercept and slopeForm an equation of a parallel or perpendicular lineSolve a pair of simultaneous linear equationsDetermine parameters in a polynomial identity5C Tangents, normals and derivative conditions: Solve a linear equationDifferentiate natural exponential functions and solve simple derivative conditionsExponential growth and decay rate models5D Exponential derivatives and rate models: Simplify and combine algebraic fractions5D Exponential derivatives and rate models: Round to decimal places and significant figuresDifferentiate sine, cosine and tangent functions and solve simple derivative conditionsGeometric optimisation with a trigonometric modelSinusoidal models and their rates of changeSolve a practical right-angled-triangle problemPerimeter and area of standard shapes5E Trigonometric differentiation and rates: Round to decimal places and significant figuresApproximate a cubic root using Newton's method