Chain Rule Practice Problems
Name the outer and inner functions before differentiating.
Differentiate the outside, keep the inside, then multiply by the inside derivative.
Count the layers and confirm that each nontrivial layer contributed a factor.
The chain rule is easiest to learn by recognizing structure rather than memorizing isolated answers. If a function is built by placing one function inside another, write it as \(f(g(x))\). Its derivative is \(f'(g(x))g'(x)\). The inner expression stays intact while the outer function is differentiated, and the derivative of the inner expression becomes a multiplying factor.
Warm-up: powers of linear expressions
Differentiate \((3x+1)^5\)
The outer function is \(u^5\), and the inner function is \(u=3x+1\). Differentiate the outside to get \(5(3x+1)^4\). Multiply by the inner derivative, 3:
\[\frac{d}{dx}(3x+1)^5=15(3x+1)^4.\]
Trigonometric composite
Differentiate \(\cos(x^2)\)
The outer derivative is \(-\sin(u)\). Preserve \(u=x^2\), then multiply by \(u'=2x\):
\[\frac{d}{dx}\cos(x^2)=-2x\sin(x^2).\]
The negative sign comes from cosine; the \(2x\) comes from the inner square.
Exponential composite
Differentiate \(e^{4x-7}\)
The derivative of \(e^u\) is \(e^u\). The inner derivative of \(4x-7\) is 4, so the result is \(4e^{4x-7}\). The exponent is preserved; it is not reduced by one.
Logarithmic composite and its domain
Differentiate \(\ln(x^2+1)\)
The derivative of \(\ln(u)\) is \(u'/u\). Here \(u'=2x\), giving \(2x/(x^2+1)\). Because \(x^2+1>0\) for every real \(x\), the original logarithm and its derivative are defined everywhere.
A three-layer problem
For \(y=\sin^3(2x)\), read the layers from outside to inside: cube, sine, and \(2x\). Differentiate each layer in order and multiply:
\[y'=3\sin^2(2x)\cdot\cos(2x)\cdot2\]
After multiplying the constants, \(y'=6\sin^2(2x)\cos(2x)\).
A reliable check is to count the three derivative contributions: 3 from the cube, cosine from sine, and 2 from the inner linear expression.
How to grade your own answer
Equivalent forms may look different. A factored answer and an expanded answer can both be correct, so compare structure before comparing typography. First verify the outer derivative, then locate the inner derivative as a factor. Substitute a safe number such as \(x=0\) only as a quick check; agreement at one point cannot prove two formulas are identical. The calculator's numerical badge samples several safe points, but the written rule remains the mathematical justification.
Diagnostic checklist
- Did the inside expression remain unchanged inside the outer derivative?
- Did every nonconstant inner layer contribute a derivative factor?
- Did you preserve signs, especially when differentiating cosine?
- Did you state restrictions for logarithms, roots, or denominators?
- Can the final expression be factored without changing its meaning?
Independent practice
- Differentiate \((5x-2)^7\).
Show solution
Outer derivative: \(7(5x-2)^6\). Inner derivative: 5. Answer: \(35(5x-2)^6\).
- Differentiate \(\sin(4x^2)\).
Show solution
Outer derivative: \(\cos(4x^2)\). Inner derivative: \(8x\). Answer: \(8x\cos(4x^2)\).
- Differentiate \(\sqrt{1+x^2}\).
Show solution
Rewrite as \((1+x^2)^{1/2}\). The derivative is \(\tfrac12(1+x^2)^{-1/2}(2x)=x/\sqrt{1+x^2}\).
- Differentiate \(e^{\sin x}\).
Show solution
The exponential stays unchanged and multiplies the derivative of \(\sin x\): \(e^{\sin x}\cos x\).
Review the full chain rule lesson, then compare the dedicated walkthroughs for sin(2x), e^(x²), and ln(2x).
Reference standard: Rule statements and notation follow OpenStax Calculus Volume 1, Chapter 3. DerivCalc's explanations and examples are independently written.