Study guide
Once the derivative is defined, the exam expects fluency in computing it quickly for combinations of functions, then applying that computation to real situations involving motion, changing quantities, and comparative growth rates. This chapter moves from mechanical rules to the word problems that dress those rules in physical meaning.
The Chain Rule and Implicit Differentiation
The chain rule handles composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) times g'(x) — differentiate the outer function while leaving the inner function alone, then multiply by the derivative of the inner function. For y = sin(x^3), the outer function is sine and the inner is x^3, so dy/dx = cos(x^3) times 3x^2, written as 3x^2 cos(x^3). The chain rule stacks for nested compositions, and most exam derivatives combine it with the product or quotient rule, so practicing multi-step differentiation is essential. Implicit differentiation applies the chain rule to equations where y is not isolated, such as x^2 + y^2 = 25, the equation of a circle. Differentiate both sides with respect to x, treating y as a function of x and multiplying every derivative of a y-term by dy/dx: 2x + 2y(dy/dx) = 0. Solving gives dy/dx = -x/y. At the point (3, 4) on this circle, the slope of the tangent line is -3/4. Implicit differentiation is indispensable whenever a curve is defined by an equation that mixes x and y terms in a way that cannot easily be solved for y alone, such as x^3 + y^3 = 6xy, and it also underlies related rates problems, since both sides of an equation are frequently differentiated with respect to time.
Derivatives of Inverse Functions and Inverse Trig Functions
If g is the inverse of a differentiable function f, then g'(x) = 1/f'(g(x)), provided f'(g(x)) does not equal zero; this formula comes from differentiating both sides of f(g(x)) = x with the chain rule. In practice, exam questions often give you a table of values for f and f' at specific points and ask for the derivative of the inverse at a corresponding point, so recognizing which input-output pair to use is the main challenge, not the algebra. The inverse trigonometric derivatives are worth memorizing outright: the derivative of arcsin(x) is 1/sqrt(1 - x^2), the derivative of arccos(x) is -1/sqrt(1 - x^2), and the derivative of arctan(x) is 1/(1 + x^2). These combine with the chain rule just like any other derivative — for instance, the derivative of arctan(3x) is 1/(1 + 9x^2) times 3, or 3/(1 + 9x^2). A helpful memory anchor is that the 'co' functions (arccos and arccot) carry the negative sign, mirroring the pattern from ordinary trig derivatives. Because these formulas appear less often than polynomial or exponential rules, students frequently under-practice them, making them a reliable source of a few recoverable points if reviewed deliberately.
Related Rates
Related rates problems connect the rates of change of two or more quantities that are linked by an equation, using implicit differentiation with respect to time t. The reliable strategy is: draw and label a diagram if one is not given, write an equation relating the variables, differentiate both sides with respect to t, substitute known values only after differentiating, and solve for the unknown rate. Consider a spherical balloon whose volume is V = (4/3) pi r^3. If the radius is 3 centimeters and growing at 2 centimeters per minute, differentiating gives dV/dt = 4 pi r^2 (dr/dt). Substituting r = 3 and dr/dt = 2 yields dV/dt = 4 pi (9)(2) = 72 pi, approximately 226.19 cubic centimeters per minute. Common related-rates scenarios include a ladder sliding down a wall (Pythagorean theorem relates the two legs), shadows lengthening as a person walks away from a lamppost (similar triangles), and water draining from a conical tank (volume of a cone with radius proportional to height). The most frequent error is substituting numeric values before differentiating, which freezes a variable that should still be changing — always differentiate the general relationship first.
Rectilinear Motion and L'Hopital's Rule
For an object moving along a line with position function s(t), velocity is v(t) = s'(t) and acceleration is a(t) = v'(t) = s''(t). The object is speeding up when velocity and acceleration share the same sign, and slowing down when they have opposite signs; the object is momentarily at rest when v(t) = 0. For s(t) = t^3 - 6t^2 + 9t, velocity is v(t) = 3t^2 - 12t + 9 = 3(t-1)(t-3), which equals zero at t = 1 and t = 3, marking the moments the particle changes direction. Total distance traveled over an interval requires splitting at these zeros and summing the absolute value of displacement on each sub-interval, since simply evaluating s(end) - s(start) only gives net displacement. L'Hopital's Rule addresses limits that produce the indeterminate forms 0/0 or infinity/infinity: if lim(x to a) f(x)/g(x) has one of these forms, then the limit equals lim(x to a) f'(x)/g'(x), provided that second limit exists. For lim(x to 0) of (e^x - 1)/x, direct substitution gives 0/0, so differentiating numerator and denominator separately gives lim(x to 0) of e^x/1 = 1. L'Hopital's Rule can be applied repeatedly if the new limit is still indeterminate, but you must re-verify the indeterminate form each time — applying it to a limit that already evaluates to a real number produces a wrong answer.
Key terms
- Chain rule
- — The differentiation rule for composite functions: the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x).
- Implicit differentiation
- — Differentiating both sides of an equation relating x and y with respect to x, applying the chain rule to every y-term and solving for dy/dx.
- Inverse function derivative formula
- — For g the inverse of f, g'(x) equals 1 divided by f'(g(x)), derived by differentiating f(g(x)) = x.
- Related rates
- — Problems that find the rate of change of one quantity using the known rate of change of a related quantity, via implicit differentiation with respect to time.
- Velocity
- — The derivative of position with respect to time, s'(t), representing both speed and direction of motion.
- Acceleration
- — The derivative of velocity with respect to time, v'(t), equivalently the second derivative of position.
- Speeding up / slowing down
- — An object speeds up when velocity and acceleration have the same sign, and slows down when they have opposite signs.
- L'Hopital's Rule
- — A technique for evaluating indeterminate limits of the form 0/0 or infinity/infinity by differentiating the numerator and denominator separately.
- Indeterminate form
- — An expression like 0/0 or infinity/infinity that does not immediately reveal the value of a limit and requires further technique.
- Total distance traveled
- — The sum of the absolute values of displacement over each interval where velocity does not change sign, distinct from net displacement.
Exam tips
- In related rates problems, write the general equation and differentiate with respect to time before plugging in any numbers — substituting too early is the most common error.
- Memorize all three inverse trig derivative formulas cold; they rarely get partial credit if only the pattern is half-remembered.
- Distinguish net displacement from total distance traveled by checking whether velocity changes sign on the interval in question.
- Before applying L'Hopital's Rule, always confirm the limit is truly in an indeterminate form (0/0 or infinity/infinity) by substitution first.
- When asked for the derivative of an inverse function at a point, locate the matching input value for the original function from any given table before applying the formula.