Chain Rule with Trigonometric Functions (3.1.5) | AP Calculus AB Notes

AP Syllabus focus:
‘Differentiate composite trigonometric functions such as sin(3x), cos(g(x)), or tan(2x − 1) by applying the chain rule to both the trig function and the inside function.’

Composite trigonometric functions require understanding how an inner expression affects an outer trigonometric function. The chain rule provides the essential structure for differentiating these layered relationships accurately and efficiently.

Chain Rule with Trigonometric Functions

Differentiating composite trigonometric expressions is a central skill in AP Calculus AB, and it rests on applying the chain rule, which ensures both the outer trigonometric function and the inner function are properly accounted for. A composite function in this context has the form trig(inner expression), where the trigonometric function serves as the outer layer and an algebraic expression, another trig function, or a more complex differentiable function serves as the inner layer.

Composite Function: A function formed by applying one function to the result of another, written as $f(g(x))$ , where $g(x)$ is the inner function and $f$ is the outer function.

Recognizing this layered structure allows you to correctly apply the chain rule whenever trigonometric expressions involve anything more complicated than a single variable $x$ .

Identifying Outer and Inner Functions in Trigonometric Composites

Accurate differentiation begins with identifying which portion of the expression is acting as the outer function (the trigonometric function itself) and which part is the inner function (the input to the trig function).
Common forms include:

$\sin(ax + b)$ where $\sin(\cdot)$ is outer and $(ax + b)$ is inner
$\cos(g(x))$ where $\cos(\cdot)$ is outer and $g(x)$ is inner
$\tan(2x - 1)$ where $\tan(\cdot)$ is outer and $(2x - 1)$ is inner

Once the inner and outer roles are clear, the chain rule becomes straightforward to apply.

This graph compares $y=\sin(x)$ , $y=\sin(2x)$ , and $y=\sin(x/2)$ , illustrating how changing the inner coefficient alters oscillation frequency. It visually demonstrates horizontal compression and stretch produced by the inner function. These geometric changes align with the inner derivative appearing in the chain rule. Source.

The Chain Rule Applied to Trigonometric Functions

The chain rule states that to differentiate a composite function, you must differentiate the outer function first, keeping the inner expression unchanged, and then multiply by the derivative of the inner function. This process is essential when trigonometric expressions contain transformations such as horizontal stretches, compressions, or shifts.

$\text{Chain Rule: } \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$
$f(g(x))$ = Composite function
$f'(g(x))$ = Derivative of the outer function evaluated at the inner function
$g'(x)$ = Derivative of the inner function

This equation confirms that both layers contribute to the final derivative, reinforcing the importance of identifying each part clearly before differentiating.

The differentiation of trigonometric functions follows standard rules, but after applying those rules to the outer function, the chain rule requires an additional multiplication step involving the inner derivative. This mandatory step accounts for the rate at which the inner function changes and ensures the derivative accurately reflects the behavior of the composite function.

Trigonometric Derivatives Required for Chain Rule Application

To apply the chain rule effectively, students must recall the basic derivative formulas for the six primary trigonometric functions. Using these as outer derivatives, the chain rule attaches the necessary inner derivative:

$\sin(\cdot)$ differentiates to $\cos(\cdot)$
$\cos(\cdot)$ differentiates to $-\sin(\cdot)$
$\tan(\cdot)$ differentiates to $\sec^2(\cdot)$
$\cot(\cdot)$ differentiates to $-\csc^2(\cdot)$
$\sec(\cdot)$ differentiates to $\sec(\cdot)\tan(\cdot)$
$\csc(\cdot)$ differentiates to $-\csc(\cdot)\cot(\cdot)$

These formulas alone, however, apply only to simple trigonometric functions.

This panel displays the basic graphs of $\sin x$ , $\cos x$ , $\tan x$ , $\csc x$ , $\sec x$ , and $\cot x$ , illustrating their periodic shapes and key features. Viewing them together reinforces the derivative relationships described in the notes. The inclusion of asymptotes provides additional context beyond the immediate syllabus focus. Source.

Structure and Strategy for Differentiating Composite Trigonometric Functions

A systematic approach helps ensure accuracy when applying the chain rule:

Identify the outer trigonometric function before differentiating.
Hold the inner function intact while applying the trig derivative rule.
Differentiate the inner function separately, noting whether it is algebraic, exponential, logarithmic, or trigonometric.
Multiply the results to complete the chain rule process.

This structured method prevents errors such as neglecting the inner derivative, which is one of the most common mistakes in early differentiation work.

Multiple-Layer Trigonometric Composites

Some expressions contain more than one layer of composition, such as trigonometric functions applied to other trigonometric or composite expressions. In such cases, the chain rule may need to be applied repeatedly as each layer is differentiated. The process remains the same, but it is repeated as necessary:

Determine the current outermost layer.
Differentiate that layer.
Multiply by the derivative of the remaining inner portion.
Continue until all nested layers have been addressed.

Careful tracking of each layer ensures accurate differentiation and reinforces the flexibility and power of the chain rule in handling increasingly complex trigonometric expressions.

FAQ

Look for the final operation applied to x before the trigonometric function acts on it. This final operation forms the inner function.

If multiple operations occur, group them from the inside out.
• For cos(3x + 2), the inner function is 3x + 2.
• For sin((x² − 4)³), the inner function for the trig function is (x² − 4)³, even though this expression itself contains layers.

Practising this decomposition helps you avoid missing derivative factors.

The inner derivative controls how quickly the input to the trigonometric function changes. Omitting it means you are assuming the inner function changes at the same rate as x, which is rarely true.

In trigonometric functions, even small changes in the input can significantly affect oscillation, so missing the inner multiplier distorts the rate of change of the entire composite function.

Yes. Several forms almost always require the chain rule:
• Trig functions applied to polynomials, such as sin(x³ − x).
• Trig functions applied to scaled or shifted arguments, like tan(5x − 7).
• Trig functions applied to other trigonometric expressions, such as cos(sin x).

Whenever the input to the trig function is not simply x, the chain rule should be considered the default.

First, recall which trig derivatives introduce negative signs: cosine, cotangent and cosecant.

Then apply a structured method:
• Differentiate the outer trig function first, noting the correct sign.
• Leave the inner function untouched until the next step.
• Multiply by the inner derivative only after confirming the sign of the outer derivative.

Writing an intermediate line before simplifying helps prevent accidental sign changes.

A reliable approach is to label layers explicitly before differentiating.

Try the following steps:
• Identify the outermost trig function and differentiate it first.
• Treat everything inside it as a single block until the next step.
• Repeat this process inward, differentiating one layer at a time.

This prevents skipping layers and keeps track of each required inner derivative, especially in expressions like sin((3x − 1)²) or tan(sin(4x)).

Practice Questions

Question 1 (1–3 marks)
Differentiate the function f(x) = tan(4x − 3).

Question 1
Differentiate f(x) = tan(4x − 3).

• 1 mark for differentiating tan(u) correctly to sec²(u).
• 1 mark for multiplying by the derivative of the inner function 4x − 3.
• 1 mark for the correct final answer: f′(x) = 4 sec²(4x − 3).

Question 2 (4–6 marks)
Let h(x) = sin(g(x)), where g is a differentiable function.
(a) Write an expression for h′(x) in terms of g(x) and g′(x).
(b) Given g(2) = 5 and g′(2) = −3, find h′(2).
(c) Explain why the chain rule is required to differentiate h(x).

Question 2

(a)
• 1 mark for recognising the outer function as sine.
• 1 mark for a correct derivative expression: h′(x) = cos(g(x)) g′(x).

(b)
• 1 mark for correct substitution of g(2) and g′(2).
• 1 mark for correct evaluation: h′(2) = cos(5) × (−3).

(c)
• 1 mark for a clear explanation that h(x) is a composite function with an inner function g(x), so the chain rule is required to differentiate the outer sine function and then multiply by the derivative of the inner function.

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Written by:

Dr Rahil Sachak-Patwa

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.