AP Calculus BC / Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
3.6 Calculating Higher-Order Derivatives
Notes
The second derivative $f''(x)$ is the derivative of $f'(x)$. Notation: $f''$, $y''$, $\displaystyle \frac{d^2y}{dx^2}$.
- Position → velocity → acceleration: $s$, $s'$, $s''$.
- For implicit equations, substitute $\displaystyle \frac{dy}{dx}$ back in.
Practice
$f(x)=x^4$. Find $f''(x)$.
$f'=4x^3$, $f''=12x^2$.
$\displaystyle \frac{d^2}{dx^2}\sin x=$
$\cos x$, then $-\sin x$.
If $s(t)=t^3-6t$, the acceleration at $t=2$ is:
$s''(t)=6t$.
Ready for more? Take the Unit 3 test.