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Chain Rule Worked Examples cheat sheet - grade 11-12

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Calculus Grade 11-12

Chain Rule Worked Examples Cheat Sheet

A printable reference covering the chain rule, composite functions, power, trigonometric, exponential, and logarithmic derivatives for grades 11-12.

TopicsDerivativesFunctionsTrigonometryExponentsCalculusAlgebra
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The chain rule is used to differentiate composite functions, where one function is placed inside another. This cheat sheet helps students recognize inside and outside functions and apply the rule step by step. Worked examples are especially useful because chain rule problems often look different even when they follow the same pattern.

Students in grades 11-12 need this reference for derivatives involving powers, radicals, trigonometric functions, exponentials, and logarithms.

The core idea is to differentiate the outside function while leaving the inside function unchanged, then multiply by the derivative of the inside function. In function notation, if y=f(g(x))y = f(g(x)), then dydx=f′(g(x))g′(x)\frac{dy}{dx} = f'(g(x))g'(x). In Leibniz notation, if y=f(u)y = f(u) and u=g(x)u = g(x), then dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.

The most important skill is identifying the inner function uu before differentiating.

Key Facts

  • The chain rule states that if y=f(g(x))y = f(g(x)), then dydx=f′(g(x))g′(x)\frac{dy}{dx} = f'(g(x))g'(x).
  • Using substitution, if u=g(x)u = g(x) and y=f(u)y = f(u), then dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.
  • For a power of a function, ddx[un]=nun−1dudx\frac{d}{dx}[u^n] = n u^{n-1}\frac{du}{dx}.
  • For a radical expression, rewrite first when helpful, such as u=u12\sqrt{u} = u^{\frac{1}{2}}, then use ddx[u12]=12u−12dudx\frac{d}{dx}[u^{\frac{1}{2}}] = \frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}.
  • For a sine composite, ddx[sin⁡(u)]=cos⁡(u)dudx\frac{d}{dx}[\sin(u)] = \cos(u)\frac{du}{dx}.
  • For a cosine composite, ddx[cos⁡(u)]=−sin⁡(u)dudx\frac{d}{dx}[\cos(u)] = -\sin(u)\frac{du}{dx}.
  • For an exponential composite, ddx[eu]=eududx\frac{d}{dx}[e^u] = e^u\frac{du}{dx}.
  • For a logarithmic composite, ddx[ln⁡(u)]=1ududx\frac{d}{dx}[\ln(u)] = \frac{1}{u}\frac{du}{dx}.

Vocabulary

Chain rule
A derivative rule used for composite functions, written as ddx[f(g(x))]=f′(g(x))g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x).
Composite function
A function made by putting one function inside another, such as f(g(x))f(g(x)).
Inner function
The inside expression in a composite function, often labeled u=g(x)u = g(x).
Outer function
The function applied to the inner expression, such as f(u)f(u) when y=f(g(x))y = f(g(x)).
Derivative
A function that gives the instantaneous rate of change of another function, often written as dydx\frac{dy}{dx}.
Leibniz notation
A notation for derivatives that shows related variables, such as dydx=dydu⋅dudx\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.

Common Mistakes to Avoid

  • Forgetting to multiply by the inner derivative is wrong because the chain rule requires the factor g′(x)g'(x) in ddx[f(g(x))]=f′(g(x))g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x).
  • Differentiating the inner function first and stopping is wrong because the outside function must also be differentiated, as in ddx[(3x+1)5]=5(3x+1)4⋅3\frac{d}{dx}[(3x+1)^5] = 5(3x+1)^4\cdot 3.
  • Changing the inside expression while differentiating the outside is wrong because the outside derivative keeps the inner expression unchanged, such as ddx[sin⁡(x2)]=cos⁡(x2)⋅2x\frac{d}{dx}[\sin(x^2)] = \cos(x^2)\cdot 2x.
  • Dropping a negative sign in trigonometric derivatives is wrong because ddx[cos⁡(u)]=−sin⁡(u)dudx\frac{d}{dx}[\cos(u)] = -\sin(u)\frac{du}{dx}.
  • Treating ln⁡(u)\ln(u) like 1x\frac{1}{x} is wrong because the derivative is 1ududx\frac{1}{u}\frac{du}{dx}, not just 1x\frac{1}{x}.

Practice Questions

  1. 1 Differentiate y=(4x2−3x+1)6y = (4x^2 - 3x + 1)^6.
  2. 2 Find dydx\frac{dy}{dx} for y=sin⁡(5x3−2x)y = \sin(5x^3 - 2x).
  3. 3 Differentiate y=exy = e^{\sqrt{x}}.
  4. 4 Explain why ddx[ln⁡(2x2+1)]\frac{d}{dx}[\ln(2x^2 + 1)] needs the chain rule, and identify the inner and outer functions.

Understanding Chain Rule Worked Examples

A useful way to think about a composite expression is as a chain of operations. Start with a number x. One operation changes it, then another operation uses that result.

For example, in the cube of three x plus one, the first operation is multiply x by three and add one. The second operation is cubing. A small change in x is enlarged by the first operation before the cube responds to it.

The derivative must account for both rates of change. This is why a missing inner derivative gives an answer that has the right general shape but the wrong size or sign.

Worked examples become easier when students mark the layers before doing any differentiation. In the expression sine of x squared minus four, the outer layer is sine. The next layer is the square.

The deepest layer is x squared minus four. Work from the outside inward, then include a derivative factor for each inner layer. This habit is especially important when several functions are nested.

A common error is to stop after differentiating sine, or to differentiate x squared minus four but forget the cosine factor. Writing each layer on a separate line can prevent both errors.

The chain rule often appears together with other derivative rules. For example, a product may contain an exponential whose exponent is a polynomial. The product rule handles the multiplication between the large pieces.

The chain rule handles the changing exponent inside the exponential. In a quotient, the quotient rule may be needed first, while each part can still require its own chain rule step. Students should identify the main structure before choosing a rule.

Ask whether the expression is built by addition, multiplication, division, or one function wrapped around another. More than one answer can apply, but the outermost structure usually determines the first rule.

These derivatives matter whenever a quantity depends on another changing quantity. In physics, position can depend on time through a squared or trigonometric expression. Differentiating gives velocity, and a chain rule factor can represent how quickly the input itself changes with time.

In biology, a population model may use an exponential expression with a time dependent rate. In economics, a cost or revenue formula can include logarithms or powers of changing variables. When checking an answer, look for three things.

Keep the original inside expression where the outer derivative needs it. Include every inner derivative factor.

Finally, test simple features such as signs and constants. If the inside function decreases, its negative derivative should affect the final result.