Introduction
A trigonometric derivative can look familiar and still be wrong. For example, differentiating \(\sin(3x)\) by changing only sine to cosine gives \(\cos(3x)\), but this leaves out a factor. That missing factor matters because the angle \(3x\) changes three times as fast as \(x\).
This is a common instance of the chain rule: when a function is composed with another function, its derivative includes the rate of change of the inner function. In trigonometric differentiation, the inner function is the angle supplied to sine or cosine.
The Chain Factor in Sine and Cosine Derivatives
Throughout, trigonometric arguments are measured in radians. The standard derivatives are \(\frac{d}{du}\sin u=\cos u\) and \(\frac{d}{du}\cos u=-\sin u\). If the angle is itself a differentiable function \(g(x)\), the chain rule gives:
The formula has two parts. The derivative of the outer function changes sine to cosine, or cosine to negative sine. The inner function remains in the trigonometric expression. Finally, multiply by the derivative of that inner function.
For \(\sin(3x)\), the outer function is sine and the inner function is \(g(x)=3x\). Since \(g'(x)=3\), the derivative is \(3\cos(3x)\), not merely \(\cos(3x)\). For \(\cos(x^2)\), the inner function is \(g(x)=x^2\), so the derivative includes \(2x\).
Why the Inner Derivative Appears
The chain factor is not an extra convention to memorize. It records how quickly the angle changes when \(x\) changes. A short limit argument shows why it must be present.
Let \(a=g(x)\), and let \(\delta=g(x+h)-g(x)\). The change in the sine value can be written using a trigonometric identity:
When \(\delta=0\), interpret the middle factor as \(1\); both sides of the identity are then \(0\). Because \(g\) is differentiable, \(\delta\) tends to \(0\) as \(h\) tends to \(0\), and \(\delta/h\) tends to \(g'(x)\). The standard limit \(\lim_{u\to 0}\sin(u)/u=1\) and continuity of cosine therefore give:
For cosine, use \(\cos(a+\delta)-\cos(a)=-2\sin(a+\delta/2)\sin(\delta/2)\). The same limit reasoning gives:
In both cases, the change in the inner angle contributes the factor \(\delta/h\), whose limit is \(g'(x)\). Leaving out that factor ignores how rapidly the angle changes.
Worked Example: Differentiating \(\sin(3x)\)
Worked Example: Differentiating \(\sin(3x)\)
Find the derivative of \(f(x)=\sin(3x)\), and compare it with the result of omitting the chain factor.
Step 1: Identify the inner function. The angle is \(g(x)=3x\), so \(g'(x)=3\).
Step 2: Differentiate the outer function. The derivative of sine is cosine, leaving the angle \(3x\) in place.
Step 3: Check by a difference quotient. Put \(\delta=3h\) in the sine difference identity:
As \(h\to 0\), the cosine factor tends to \(\cos(3x)\), and the sine ratio tends to \(1\). The limit is \(3\cos(3x)\), confirming the derivative.
Step 4: See why the omission matters. At \(x=0\), the correct derivative is \(3\cos(0)=3\), whereas the expression with the factor omitted gives \(\cos(0)=1\). They disagree. But at \(x=\pi/6\), the angle is \(3x=\pi/2\), so both expressions equal \(0\). Agreement at that one point does not make the omitted-factor formula correct.
Worked Example: Differentiating \(\cos(x^2)\)
Worked Example: Differentiating \(\cos(x^2)\)
Find the derivative of \(f(x)=\cos(x^2)\). Here the angle changes with \(x\), but not at a constant rate.
Step 1: Identify the inner function and its derivative.
Step 2: Apply the cosine chain rule. The outer derivative is negative sine, and the chain factor is \(2x\):
Step 3: Check with the difference quotient. For an increment \(h\), the change in the inner function is \(\delta=(x+h)^2-x^2=2xh+h^2=h(2x+h)\). The cosine difference identity gives:
As \(h\to 0\), \(\delta\to 0\), the sine ratio tends to \(1\), and \(2x+h\) tends to \(2x\). The limit is \(-2x\sin(x^2)\), as found by the chain rule.
Step 4: Compare at \(x=1\). The correct derivative is \(-2\sin(1)\). Omitting the chain factor gives \(-\sin(1)\), which is different because \(\sin(1)\ne 0\).
Worked Example: A Shifted Linear Angle
Worked Example: A Shifted Linear Angle
Differentiate \(f(x)=\cos(3x+\pi/4)\), then evaluate the derivative at \(x=\pi/12\).
Step 1: Differentiate the inner angle.
Step 2: Differentiate the outer cosine and include the chain factor.
Step 3: Evaluate at the specified input.
Step 4: Check by a difference quotient. Replacing \(x\) by \(x+h\) changes the angle by \(3h\). The cosine difference identity yields a quotient whose limit is \(-3\sin(3x+\pi/4)\), since \(\sin(3h/2)/(3h/2)\to 1\). At \(x=\pi/12\), that limit is \(-3\sin(\pi/2)=-3\), matching the evaluated derivative.
If the factor \(3\) were omitted, the derivative at this input would be \(-1\), not \(-3\).
Diagnosing the Missing-Factor Error
A common incorrect calculation is:
This calculation differentiates the outer cosine but stops before differentiating its angle. Correct the calculation by identifying the inner function \(g(x)=x^2\) and multiplying by \(g'(x)=2x\):
An omitted factor does not necessarily cause the two expressions to differ at every input. If \(g'(x)=1\), the factor does not change the formula. Omitting it generally gives an incorrect derivative formula, though the two expressions can happen to agree at points where the outer derivative is zero. For instance, in the first example, both expressions vanish when \(3x=\pi/2\). This pointwise agreement does not validate the incomplete formula for other inputs.
A reliable check is to name the inner angle before differentiating. Then ask: Have I multiplied by the derivative of that angle? This simple check catches omissions in both linear angles, such as \(3x+\pi/4\), and nonlinear angles, such as \(x^2\).
Why the Chain Factor Matters
The derivative measures change with respect to \(x\). A trigonometric value can change quickly not only because of the behavior of sine or cosine, but also because its angle changes quickly. The chain factor accounts for this second rate of change.
For example, in \(\sin(3x)\), the input to sine changes three times as quickly as \(x\). In \(\cos(x^2)\), the angle changes at rate \(2x\), which varies with \(x\). Leaving out that rate loses information about the composite function's slope.
When differentiating a sine or cosine composite, use this sequence:
Write the inner function \(g(x)\) explicitly.
Sine becomes cosine; cosine becomes negative sine.
Include the inner derivative \(g'(x)\).
Test a suitable input or use the difference quotient reasoning to see whether an omitted factor would change the value.
The essential distinction is between differentiating the outer trigonometric function and differentiating the full composite. The chain factor completes the second task.
Check Your Understanding
For each question, identify the inner function before deciding whether the derivative includes a chain factor.
- Differentiate \(\sin(5x)\). What is the inner function and its derivative?
- Differentiate \(\cos(2x+\pi/3)\), and evaluate the derivative at \(x=0\).
- Differentiate \(\sin(x^2+1)\). Which factor would be missing in the calculation \(\cos(x^2+1)\)?
- For \(\sin(3x)\), why can the correct and incomplete expressions agree at \(x=\pi/6\), even though the incomplete derivative formula is wrong?
- State the chain rule formula for \(\cos(g(x))\), including the condition on \(g\).
