Calculus I Foundations (MATH 151) · Module 4 of 11
The Chain Rule, Implicit Differentiation & Higher-Order Derivatives
Differentiating functions built from other functions, and equations that never isolate y.
The Chain Rule
- The chain rule differentiates a composite function: d/dx[f(g(x))] = f'(g(x))·g'(x) — "derivative of the outside, leaving the inside alone, times the derivative of the inside.".
Find the derivative of f(x) = (3x+1)⁵ using the chain rule.
Solution. f'(x) = 5(3x+1)⁴ · 3 = 15(3x+1)⁴.
Combining the Chain Rule with Other Rules
- The chain rule often combines with the product or quotient rule when a composite function is multiplied or divided by another function.
Find the derivative of f(x) = x²·sin(3x), which requires both the product rule and the chain rule.
Solution. f'(x) = 2x·sin(3x) + x²·cos(3x)·3 = 2x sin(3x) + 3x² cos(3x) — product rule on x² and sin(3x), with the chain rule applied to differentiate sin(3x).
Implicit Differentiation
- Implicit differentiation finds dy/dx for equations not solved for y, by differentiating both sides with respect to x, applying the chain rule to any term involving y, and then solving algebraically for dy/dx.
Find dy/dx for the equation x² + y² = 25 using implicit differentiation.
Solution. Differentiate both sides: 2x + 2y(dy/dx) = 0. Solve for dy/dx: dy/dx = −x/y.
Higher-Order Derivatives
- The second derivative f''(x) is the derivative of the derivative, measuring how the rate of change itself is changing (concavity); higher-order derivatives continue this pattern.
If f(x) represents position, what do f'(x) and f''(x) represent physically?
Solution. f'(x) represents velocity (the rate of change of position), and f''(x) represents acceleration (the rate of change of velocity).
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. What does the chain rule compute?
2. Find the derivative of (2x + 5)³ using the chain rule.