Inside This Unit: The Full Breakdown
The derivative measures instantaneous rate of change and is defined as a limit. This unit builds fluency with derivative rules — power, product, quotient — and derivatives of trigonometric, exponential, and logarithmic functions.
Why it matters
Every subsequent BC unit depends on strong differentiation skills. Speed and accuracy with these rules directly impact your performance on time-pressured multiple-choice sections.
Key concepts
- The derivative at a point is the slope of the tangent line, defined as a limit of the difference quotient.
- Power rule, product rule, and quotient rule are the workhorses of differentiation.
- Derivatives of sin, cos, tan, e^x, and ln(x) must be automatic.
- Differentiability implies continuity but not vice versa.
Definition and Interpretation
The derivative f'(a) = lim as h approaches 0 of [f(a+h) - f(a)]/h. This limit gives the slope of the tangent line to f at x = a and represents the instantaneous rate of change. The tangent line equation y = f(a) + f'(a)(x-a) serves as the best linear approximation near x = a. If the limit fails to exist due to a corner, cusp, discontinuity, or vertical tangent, the function is not differentiable at that point.
Derivative Rules
The power rule d/dx[x^n] = nx^(n-1) works for all real exponents, including negatives and fractions. The product rule d/dx[fg] = f'g + fg' and quotient rule d/dx[f/g] = (f'g - fg')/g^2 handle combinations. The constant multiple and sum rules allow term-by-term differentiation. These rules combine freely, and BC students should be able to differentiate any algebraic expression quickly and correctly without hesitation.
Transcendental Derivatives
Memorize these: d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[e^x] = e^x, d/dx[ln x] = 1/x, d/dx[a^x] = a^x ln a, d/dx[log_a x] = 1/(x ln a). These appear constantly in BC-specific contexts like series for e^x and sin x. The chain rule (covered next unit) multiplies these by the derivative of the inner function, but first ensure the base derivatives are second nature.
AP exam tip
The AP exam sometimes gives a table of function values and asks for the derivative of a product or quotient at a specific point. Set up the product or quotient rule formula first, then substitute values from the table.
Connections to other units
- Unit 3: The chain rule extends these basic rules to compositions of functions.
- Unit 10: Taylor series use repeated differentiation to build polynomial approximations.
- Unit 6: Every derivative rule has a corresponding antiderivative rule used in integration.
