5.4 Derivatives and Applications

Why This Topic Matters

The official differential-calculus competency names the derivative as tangent slope and as the limit of a difference quotient, then adds derivative calculations, graph analysis, rates of change, and optimization. That list signals a broad exam target. You are not only finding f'(x); you are deciding what f'(x) means in a graph, table, or application.

The derivative of f at x = a is the instantaneous rate of change of f at that input. Geometrically, it is the slope of the tangent line. If f'(a) is positive, the function is increasing at that point; if f'(a) is negative, it is decreasing. If f'(a) = 0, the tangent is horizontal, but that alone does not guarantee a maximum or minimum.

Rules And Recognition

For exponential and logarithmic functions, know the common rules: d/dx e^x = e^x, d/dx ln x = 1/x for x > 0, and chain-rule versions such as d/dx ln(5x + 2) = 5/(5x + 2). For trig functions, know at least d/dx sin x = cos x and d/dx cos x = -sin x.

Worked computation: find the tangent line to f(x) = x^3 - 4x + 1 at x = 2. First, f(2) = 8 - 8 + 1 = 1. Next, f'(x) = 3x^2 - 4, so f'(2) = 8. The tangent line has slope 8 through (2, 1): y - 1 = 8(x - 2), or y = 8x - 15. A common error is to use f(2) = 1 as the slope. The function value is a height; the derivative value is the slope.

Graph Analysis

Critical points occur where f'(x) = 0 or f'(x) is undefined, as long as x is in the domain of f. Use a sign chart for f' to determine increasing and decreasing intervals. If f' changes from positive to negative, f has a local maximum. If f' changes from negative to positive, f has a local minimum. If the sign does not change, the critical point may be a flat point without an extremum.

The second derivative adds concavity. If f''(x) > 0, the graph is concave up; if f''(x) < 0, it is concave down. Inflection points require a concavity change, not merely f''(x) = 0. In teaching terms, a student who says "second derivative equals zero, so inflection point" has skipped the sign-change test.

Optimization Routine

Optimization questions are often less about advanced calculus and more about modeling. Use this routine:

1. Define the variables and the quantity to maximize or minimize.
2. Use constraints to write that quantity as one function of one variable.
3. State the realistic domain.
4. Differentiate and find critical points.
5. Check endpoints and interpret the result in context.

Example: A rectangle has perimeter 40, so 2L + 2W = 40 and W = 20 - L. Its area is A(L) = L(20 - L) = 20L - L^2, with 0 < L < 20. Then A'(L) = 20 - 2L, so L = 10 and W = 10. The maximum area occurs for a square. This can also be justified algebraically, but derivative reasoning shows why the turning point matters.

Rates And Units

If position is in meters and time is in seconds, velocity is meters per second and acceleration is meters per second squared. If revenue is dollars and quantity is items, marginal revenue is dollars per item. AEPA can ask you to choose an interpretation; the correct answer should match both the derivative and the units.

Common traps include applying product rule to a sum, forgetting the chain-rule multiplier, finding only f'(x) when the question asks for a tangent line, and reporting a critical number without checking whether it is a maximum, minimum, or neither. Keep the function, derivative, and interpretation in separate lines of work.

Related Rates And Modeling Checks

Related-rates problems are derivative applications in disguise. Identify the variables that change with time, write a relationship among them, differentiate implicitly with respect to time, then substitute known values. If a circular oil spill has area A = pi r^2, differentiating gives dA/dt = 2pi r dr/dt. Substituting before differentiating is a common mistake because it turns a changing radius into a constant and loses the rate relationship.

Even when an item is multiple-choice, write the units beside each derivative. If r is feet and t is minutes, dr/dt is feet per minute, while dA/dt is square feet per minute. Units can reveal whether the answer choice is a radius, an area, or a rate of area change.

Calculator use should be strategic. The on-screen scientific calculator can evaluate numerical expressions, but it will not decide whether product rule, quotient rule, or chain rule applies. On a certification exam, that decision is part of the content knowledge. If a derivative result seems unreasonable, check it with a quick slope estimate from nearby function values or with the expected sign of the graph.