First-year courses · Mathematics

Calculus I: derivatives and applications

Derivative rules, chain rule, implicit differentiation, related rates, L'Hospital's rule, curve sketching, optimization and the Mean Value Theorem.

41 cards6 topicsAbout 15 minutes a roundTypical first-year Calculus I (differential calculus), derivatives and their applications

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About this deck

These cards follow the core of a first-year differential calculus course once limits are done. They cover the limit definition of the derivative, the power, product, quotient and chain rules, derivatives of trig, exponential, logarithmic and inverse trig functions, implicit and logarithmic differentiation, related rates, linear approximation, L'Hospital's rule, curve sketching with the first and second derivatives, optimization, the Mean Value Theorem and Newton's method.

You should already be able to evaluate limits, simplify algebraic fractions and work with trig identities and log laws. If the limits deck still feels shaky, review it first; the definition of the derivative is a limit and several cards rely on it.

Rules are best memorized by using them, so cover the example on each formula card and differentiate the function yourself before checking. For the application cards (related rates, optimization), write out the steps you would follow before flipping, then compare with the card. Once a tag feels easy, mix the whole deck so that you practise recognizing which rule a function needs, which is the real skill tested on exams.

Frequent errors include forgetting the inner derivative in the chain rule, writing the quotient rule numerator in the wrong order (it is f'g - fg', not fg' - f'g), and substituting numbers into a related rates problem before differentiating, which turns changing quantities into constants. Another classic: applying L'Hospital's rule to a limit that is not of the form 0/0 or infinity/infinity, or differentiating the quotient with the quotient rule instead of differentiating top and bottom separately.

Before you start: Limits and continuity from the start of Calculus I, plus Grade 12 algebra, logarithms and trigonometry.

All 41 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Definition and basic rules

Derivative of f at a: f'(a) = the limit as h approaches 0 of (f(a + h) - f(a))/h, when it exists: the instantaneous rate of change and the slope of the tangent line at a.Example: f(x) = x^2, a = 3: ((3 + h)^2 - 9)/h = 6 + h, which approaches 6.
Tangent line equation: The line through (a, f(a)) with slope f'(a): y = f(a) + f'(a)(x - a).Example: For f(x) = x^2 at a = 3: y = 9 + 6(x - 3), that is y = 6x - 9.
Differentiability and continuity: A function with a derivative at a point must be continuous there, but a continuous function can fail to have a derivative (corner, cusp, vertical tangent).Example: |x| is continuous at 0 but has a corner there: slopes -1 on the left and 1 on the right.
Power rule: For any real constant n, the derivative of x^n is n x^(n - 1).Example: d/dx x^5 = 5x^4; d/dx x^(-2) = -2x^(-3); d/dx sqrt(x) = 1/(2 sqrt(x)).
Sum and constant multiple rules: Differentiate term by term, and constants that multiply a function stay in front: (cf + g)' = cf' + g'.Example: d/dx (4x^3 - 7x + 2) = 12x^2 - 7.
Product rule: (fg)' = f'g + fg': derivative of the first times the second, plus the first times the derivative of the second.Example: d/dx (x^2 sin x) = 2x sin x + x^2 cos x.
Quotient rule: (f/g)' = (f'g - fg')/g^2 wherever g is not 0. Remember: low d-high minus high d-low, over low squared.Example: d/dx x/(x + 1) = ((x + 1) - x)/(x + 1)^2 = 1/(x + 1)^2.
Derivatives of sine and cosine: d/dx sin x = cos x and d/dx cos x = -sin x, valid when x is in radians.Example: d/dx (3 cos x) = -3 sin x.
Derivatives of tan, sec, csc and cot: tan x gives sec^2 x; sec x gives sec x tan x; csc x gives -csc x cot x; cot x gives -csc^2 x.Example: d/dx tan x at x = 0 is sec^2(0) = 1.
Exponential derivatives: e^x is its own derivative; for any base a > 0, the derivative of a^x is a^x ln a.Example: d/dx 2^x = 2^x ln 2, about 0.693 * 2^x.
Logarithm derivatives: The derivative of ln x is 1/x (for x > 0), and of log base a of x is 1/(x ln a); ln|x| also has derivative 1/x.Example: d/dx ln(5x) = 5/(5x) = 1/x.
Inverse trig derivatives: d/dx arcsin x = 1/sqrt(1 - x^2) and d/dx arctan x = 1/(1 + x^2).Example: The slope of arctan x at x = 1 is 1/2.
Higher derivatives: Differentiating again gives f'' (second derivative), f''' and so on; f'' measures how the slope itself changes.Example: Position s(t) = t^3: velocity 3t^2, acceleration 6t.

Chain rule and implicit differentiation

Chain rule: The derivative of a composition f(g(x)) is f'(g(x)) * g'(x): outer derivative evaluated at the inside, times the inside's derivative.Example: d/dx (3x + 1)^4 = 4(3x + 1)^3 * 3 = 12(3x + 1)^3.
Chain rule in Leibniz form: If y depends on u and u depends on x, then dy/dx = (dy/du)(du/dx).Example: y = sin(u), u = x^2: dy/dx = cos(x^2) * 2x.
Implicit differentiation: Differentiate both sides of an equation in x and y with respect to x, treating y as a function of x (so y^2 gives 2y dy/dx), then solve for dy/dx.Example: x^2 + y^2 = 25 gives 2x + 2y y' = 0, so y' = -x/y; at (3, 4) the slope is -3/4.
Logarithmic differentiation: Take ln of both sides, use log laws to simplify, differentiate implicitly, then multiply by y. Useful for variable exponents and messy products.Example: y = x^x: ln y = x ln x, so y'/y = ln x + 1 and y' = x^x (ln x + 1).
Derivative of an inverse function: If f(a) = b and f'(a) is not 0, the inverse function has slope 1/f'(a) at b.Example: f(x) = x^3 + x: f(1) = 2, f'(1) = 4, so the inverse has slope 1/4 at x = 2.

Rates and approximation

Related rates strategy: Draw and label, write an equation linking the quantities, differentiate with respect to time t, then substitute the known values at the instant asked.Example: Circle: A = pi r^2, dA/dt = 2 pi r dr/dt. With r = 5 cm and dr/dt = 2 cm/s, dA/dt = 20 pi cm^2/s.
Sliding ladder problem: A classic related rates set-up: a ladder of fixed length L against a wall satisfies x^2 + y^2 = L^2, so x dx/dt + y dy/dt = 0.Example: 10 m ladder, base 6 m out moving at 1 m/s: top is 8 m up and dy/dt = -(6/8)(1) = -0.75 m/s.
Linear approximation: Near x = a, f(x) is approximately L(x) = f(a) + f'(a)(x - a), the tangent line value.Example: sqrt(4.1) is about 2 + (1/4)(0.1) = 2.025; the true value is about 2.0248.
Differential dy: dy = f'(x) dx estimates the change in y caused by a small change dx in x.Example: y = x^3 at x = 2 with dx = 0.01: dy = 12(0.01) = 0.12 (true change 0.120601).
Velocity and acceleration: For position s(t), velocity is s'(t) and acceleration is s''(t); speed is |s'(t)|, and the object is at rest when s'(t) = 0.Example: s = t^2 - 4t: v = 2t - 4, at rest at t = 2; a = 2 throughout.

L'Hospital's rule

L'Hospital's rule: If f/g gives 0/0 or infinity/infinity at a, then lim f/g = lim f'/g' (top and bottom differentiated separately), provided the new limit exists or is infinite.Example: (e^x - 1)/x at 0 is 0/0; lim e^x/1 = 1.
Indeterminate product 0 times infinity: Rewrite the product as a quotient (move one factor to the denominator as a reciprocal) to get 0/0 or infinity/infinity, then apply L'Hospital.Example: x ln x as x approaches 0+: ln x/(1/x) gives (1/x)/(-1/x^2) = -x, which approaches 0.
Indeterminate powers: For forms 1^infinity, 0^0 and infinity^0, take ln of the expression, find the limit of the log, then exponentiate.Example: (1 + 1/x)^x as x approaches infinity: the log approaches 1, so the limit is e.
Checking the form before using L'Hospital: The rule only applies to 0/0 or infinity/infinity; using it on any other form gives wrong answers.Example: (x + 1)/(x + 2) at x = 0 is simply 1/2. Differentiating top and bottom would wrongly give 1.

Curve sketching

Critical number: A number c in the domain of f where f'(c) = 0 or f'(c) does not exist. Local extrema can only occur at such points (or endpoints).Example: f(x) = x^3 - 3x: f'(x) = 3x^2 - 3 = 0 at x = -1 and x = 1.
Increasing/decreasing test: If f' > 0 on an interval, f rises there; if f' < 0, f falls there.Example: For x^3 - 3x, f' < 0 on (-1, 1), so f decreases there.
First derivative test: At a critical number, if f' changes from positive to negative there is a local maximum; from negative to positive, a local minimum; no sign change, neither.Example: x^3 - 3x: f' goes + to - at x = -1, so local max f(-1) = 2.
Concavity: f is concave up where f'' > 0 (graph bends upward, slopes increasing) and concave down where f'' < 0.Example: f(x) = x^2 has f'' = 2 > 0, so it is concave up everywhere.
Inflection point: A point on the graph where concavity changes from up to down or down to up; f'' is 0 or undefined there, but that alone is not enough.Example: x^3 - 3x has f'' = 6x, which changes sign at 0, so (0, 0) is one. x^4 has f''(0) = 0 but no change.
Second derivative test: If f'(c) = 0 and f''(c) > 0, f has a local minimum at c; if f''(c) < 0, a local maximum; if f''(c) = 0 the test is inconclusive.Example: x^3 - 3x at x = 1: f''(1) = 6 > 0, so local min f(1) = -2.
Local versus absolute extrema: A local extreme value is the highest or lowest compared with nearby points; an absolute one is the highest or lowest over the whole domain considered.Example: x^3 - 3x has a local max of 2 at x = -1 but no absolute max on all real numbers.

Optimization and theorems

Extreme Value Theorem: A function that is continuous on a closed, bounded interval [a, b] attains both an absolute maximum and an absolute minimum on that interval.Example: x^2 on [-1, 2] has min 0 at x = 0 and max 4 at x = 2.
Closed interval method: To find absolute extrema of a continuous f on [a, b], evaluate f at every critical number inside and at both endpoints; the largest and smallest values win.Example: x^3 - 3x on [0, 2]: f(0) = 0, f(1) = -2, f(2) = 2. Max 2, min -2.
Optimization procedure: Name the quantity to maximize or minimize, write it as a function of one variable using the constraint, find critical numbers, and justify that the answer is a max or min.Example: Rectangle with perimeter 40 m: A = x(20 - x), A' = 20 - 2x = 0 at x = 10, giving a 10 m by 10 m square of 100 m^2.
Rolle's theorem: If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f'(c) = 0 for some c in (a, b).Example: f(x) = x^2 - 4x on [0, 4]: f(0) = f(4) = 0 and f'(2) = 0.
Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) has f'(c) = (f(b) - f(a))/(b - a): the tangent slope equals the secant slope.Example: x^2 on [1, 3]: secant slope (9 - 1)/2 = 4, and 2c = 4 at c = 2.
Zero derivative means constant: A consequence of the MVT: if f'(x) = 0 at every point of an interval, f is constant there; so two functions with equal derivatives differ by a constant.Example: Any function with derivative 2x on an interval has the form x^2 + C.
Newton's method: Root-finding iteration using tangent lines: x(n+1) = x(n) - f(x(n))/f'(x(n)). It converges fast near a simple root but can fail from a poor start.Example: f(x) = x^2 - 2, x0 = 1: x1 = 1.5, x2 = 1.5 - 0.25/3 = 1.41667 (sqrt 2 is 1.41421).