Derivatives and Their Applications
Core ideas
The derivative is a limit of a difference quotient and represents instantaneous rate of change and slope of the tangent line. Master the power, product, quotient, and chain rules, plus derivatives of $\sin x$, $\cos x$, $e^x$, and $\ln x$. Implicit differentiation handles curves not written as $y=f(x)$.
Applications drive most questions: tangent and normal lines, linear approximation, related rates, and optimization. The Mean Value Theorem guarantees a point where the instantaneous slope equals the average slope. Use the first derivative to find increasing/decreasing intervals and local extrema, and the second derivative for concavity and inflection points. Critical points occur where $f'=0$ or $f'$ is undefined.
Traps
- Forgetting the chain rule's inner derivative, especially with trig and exponential composites.
- Treating a critical point as automatically an extremum; sign changes are required.
- Mixing up the quotient rule order or dropping the minus sign.
- In optimization, ignoring endpoint candidates on a closed interval.
- Confusing concavity (second derivative) with slope (first derivative).
Worked example
Differentiate $f(x)=x^2 e^{3x}$. This is a product with a composite factor. Product rule: $f'=2x,e^{3x}+x^2\cdot\frac{d}{dx}e^{3x}$. The chain rule gives $\frac{d}{dx}e^{3x}=3e^{3x}$. So $f'=2xe^{3x}+3x^2e^{3x}=xe^{3x}(2+3x)$. Rule of thumb: identify the outermost structure first (product, quotient, or composition), then work inward.