Composite functions occur when one function is placed inside another, such as f(g(x)). Their derivatives are essential in calculus because many real formulas are built in layers, including powers of trig functions, exponentials with polynomial inputs, and square roots of expressions. The chain rule gives a reliable way to differentiate these nested structures.
It helps you track how a small change in x moves through each layer of the function.
Key Facts
- Composite function notation: y = f(g(x))
- Chain rule: d/dx[f(g(x))] = f'(g(x))g'(x)
- Leibniz form: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx)
- Power chain rule: d/dx[(g(x))^n] = n(g(x))^(n - 1)g'(x)
- Multiple layers: d/dx[f(g(h(x)))] = f'(g(h(x)))g'(h(x))h'(x)
- For y = sin(3x^2 + 1), dy/dx = cos(3x^2 + 1)(6x)
Vocabulary
- Composite function
- A function formed by using the output of one function as the input of another function.
- Inner function
- The function inside a composite expression, often written as g(x) in f(g(x)).
- Outer function
- The function applied after the inner function, often written as f in f(g(x)).
- Chain rule
- A differentiation rule that multiplies the derivative of the outer function by the derivative of the inner function.
- Intermediate variable
- A temporary variable, often u, used to represent the inner function and make the chain rule easier to apply.
Common Mistakes to Avoid
- Forgetting to multiply by the inner derivative, which is wrong because the outer function changes with its input and that input also changes with x.
- Choosing the wrong inner function, which is wrong because the chain rule depends on identifying the exact expression being substituted into the outer function.
- Differentiating f(g(x)) as f'(x)g'(x), which is wrong because the derivative of the outer function must be evaluated at g(x), not at x.
- Stopping too early in multiple nesting, which is wrong because each layer contributes its own derivative factor to the final answer.
Practice Questions
- 1 Find dy/dx for y = (5x^2 - 4x + 1)^7.
- 2 Find dy/dx for y = e^(sin(2x)) and evaluate it at x = 0.
- 3 A student says the derivative of cos(x^3 + 2) is -sin(x^3 + 2). Explain what is missing and why.