Higher-Order Derivatives and Related Rates

By the end of this lesson, you will be able to find higher-order derivatives and solve related-rates problems by differentiating equations that relate two or more variables with respect to time.

Higher-Order Derivatives

The second derivative $f''(x)$ is the derivative of the first derivative. The third derivative $f'''(x)$ is the derivative of the second, and so on.

Notation:

• Second: $f''(x)$ or $\frac{d^2y}{dx^2}$
• Third: $f'''(x)$ or $\frac{d^3y}{dx^3}$
• n-th: $f^{(n)}(x)$ or $\frac{d^ny}{dx^n}$

Example: $f(x) = x^4 - 3x^2 + 5$

$f'(x) = 4x^3 - 6x$

$f''(x) = 12x^2 - 6$

$f'''(x) = 24x$

$f^{(4)}(x) = 24$

$f^{(5)}(x) = 0$ (and all higher derivatives)

Interpretation: $f'(x)$ = velocity; $f''(x)$ = acceleration; $f''(x) > 0$ means concave up; $f''(x) < 0$ means concave down.

Introduction to Related Rates

Related rates problems involve two or more quantities that are changing with time, where a known relationship between them allows us to find the rate of change of one given the rate of change of another.

Key: Differentiate both sides of the relating equation with respect to time $t$, using the chain rule.

Strategy for Related Rates

1. Draw a picture and label variables (use letters for quantities that change)
2. Write an equation relating the variables using geometry, physics, or algebra
3. Differentiate both sides with respect to $t$
4. Substitute the known values and rates
5. Solve for the unknown rate

Example 1: Expanding Circle

A pebble dropped in a pond creates circular ripples. The radius increases at 3 cm/s. How fast is the area increasing when the radius is 10 cm?

Variables: $r$ (radius), $A$ (area), both functions of $t$.

Relating equation: $A = \pi r^2$

Differentiate with respect to t:

$$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$$

Substitute: $r = 10$, $\frac{dr}{dt} = 3$