Difficulty: Intermediate | Prerequisites: Limits and Continuity study notes, algebra of exponents and logarithms.
Differentiation is the central skill of Calculus I. Once you know how to take a derivative, you can find slopes of tangent lines, locate maxima and minima, describe how quantities change, and solve optimisation problems. This topic builds directly on limits (the derivative is itself a limit) and feeds into everything that follows, including curve sketching, related rates, and integration. If limits still feel uncertain, review those notes before diving in here.
The derivative measures the instantaneous rate of change of a function. You will need the power rule, chain rule, product rule, and quotient rule. Know the derivatives of e^x, ln(x), sin(x), and cos(x) by heart, and be able to find tangent line equations and critical points.
Derivative
The instantaneous rate of change of f(x) with respect to x, defined as f'(x) = lim(h->0) [f(x+h) - f(x)]/h. In simple terms, it tells you the slope of the curve at any given point.
Power rule
If f(x) = x^n, then f'(x) = n * x^(n-1). Works for any real constant n. Think of it as: bring the exponent down as a coefficient, then reduce the exponent by one.
Chain rule
If y = f(g(x)), then dy/dx = f'(g(x)) * g'(x). Used whenever one function is nested inside another. In simple terms, differentiate the outer function (leaving the inner alone), then multiply by the derivative of the inner function.
Product rule
If h(x) = f(x) * g(x), then h'(x) = f'(x) * g(x) + f(x) * g'(x). Think of it as: derivative of the first times the second, plus the first times the derivative of the second.
Quotient rule
If h(x) = f(x)/g(x), then h'(x) = [f'(x) * g(x) - f(x) * g'(x)] / [g(x)]^2. In simple terms, it is (low d-high minus high d-low) over (low squared).
Tangent line
The straight line that just touches the curve at a point and has the same slope as the curve there. Its equation is y - f(a) = f'(a)(x - a). This is the best linear approximation of the function near that point.
Critical point
A value x = c in the domain of f where f'(c) = 0 or f'(c) is undefined. These are the candidates for local maxima, local minima, or inflection points. Think of it as: wherever the slope is flat or the slope breaks, something interesting might happen.
d/dx [x^n] = n * x^(n-1) for any constant n.
Constants vanish: d/dx [7] = 0.
Example: g(x) = 5x^4 - 3x^2 + 7. Then g'(x) = 20x^3 - 6x. The constant 7 drops away.
Used for composite functions: d/dx [f(g(x))] = f'(g(x)) * g'(x).
Example: d/dx [sin(x^2)]. Outer function: sin(u), derivative cos(u). Inner function: u = x^2, derivative 2x. Result: 2x * cos(x^2).
Example: d/dx [(x^3 + 1)^10]. Outer: u^10, derivative 10u^9. Inner: x^3 + 1, derivative 3x^2. Result: 10(x^3 + 1)^9 * 3x^2 = 30x^2(x^3 + 1)^9.
h(x) = f(x) * g(x) gives h'(x) = f'(x)g(x) + f(x)g'(x).
Example: f(x) = x * ln(x). f'(x) = 1 * ln(x) + x * (1/x) = ln(x) + 1.
h(x) = f(x)/g(x) gives h'(x) = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2.
Example: f(x) = e^x / x. f'(x) = [e^x * x - e^x * 1] / x^2 = e^x(x - 1) / x^2.
d/dx [e^x] = e^x. The exponential function is its own derivative.
d/dx [ln(x)] = 1/x for x > 0.
d/dx [sin(x)] = cos(x).
d/dx [cos(x)] = -sin(x).
d/dx [x^e] = e * x^(e-1). Here e is treated as a constant exponent, so the power rule applies.
The slope of the tangent line to y = f(x) at x = a is f'(a).
Example: y = e^x at x = 0. Since d/dx [e^x] = e^x, the slope at x = 0 is e^0 = 1.
The equation is y - f(a) = f'(a)(x - a).
Example: f(x) = x^2 at the point (3, 9). f'(x) = 2x, so f'(3) = 6. The tangent line: y - 9 = 6(x - 3), which simplifies to y = 6x - 9.
Set f'(x) = 0 and solve.
Example: f(x) = x^2 - 4x + 5. f'(x) = 2x - 4. Setting 2x - 4 = 0 gives x = 2.
Also check where f'(x) is undefined (within the domain of f).
Power rule: d/dx [x^n] = n * x^(n-1)
Chain rule: d/dx [f(g(x))] = f'(g(x)) * g'(x)
Product rule: (fg)' = f'g + fg'
Quotient rule: (f/g)' = (f'g - fg') / g^2
d/dx [e^x] = e^x
d/dx [ln(x)] = 1/x
d/dx [sin(x)] = cos(x), d/dx [cos(x)] = -sin(x)
d/dx [x^e] = e * x^(e-1)
Tangent line at (a, f(a)): y - f(a) = f'(a)(x - a)
Derivatives measure rates of change. In physics, the derivative of position gives velocity, and the derivative of velocity gives acceleration. In economics, the marginal cost of production is the derivative of the cost function. Engineers use derivatives to find where stress on a beam is greatest, and biologists use them to model the rate of growth of a population.
Students often multiply the derivatives together for a product (writing f'(x) * g'(x) instead of using the product rule). The product rule has two terms, not one.
Students apply the quotient rule when simple algebra would suffice. If the denominator is a constant, just divide each term by it.
Students forget to multiply by the inner derivative when using the chain rule. d/dx [sin(x^2)] is 2x cos(x^2), not just cos(x^2).
Students confuse d/dx [x^e] with d/dx [e^x]. The first uses the power rule (e is a constant); the second is e^x.
The power rule, chain rule, product rule, and quotient rule are tested in almost every exam, often in combination.
"Find the tangent line" is a reliable short-answer question. Know the formula y - f(a) = f'(a)(x - a) cold.
The chain rule appears in disguise: any nested function (sin(x^2), (x^3 + 1)^10, e^(2x)) requires it.
Critical points come up in both multiple-choice ("find the critical points of...") and long-answer optimisation problems.
d/dx [e^x] = e^x and d/dx [ln(x)] = 1/x are used so often that hesitating on them costs time.
Fill in the blank: d/dx [x^7] = ___. (7x^6.)
True or False: d/dx [e^x] = x * e^(x-1). (False. d/dx [e^x] = e^x. The power rule does not apply to e^x.)
Fill in the blank: d/dx [sin(x^2)] = ___. (2x cos(x^2).)
True or False: The product rule states (fg)' = f'g'. (False. It is f'g + fg'.)
Fill in the blank: The slope of the tangent line to y = e^x at x = 0 is ___. (1.)
Q: Calculate the derivative of g(x) = 5x^4 - 3x^2 + 7.
A: Apply the power rule term by term: g'(x) = 20x^3 - 6x. The constant 7 has derivative 0.
Q: Evaluate d/dx [sin(x^2)].
A: Chain rule. Outer: sin(u) -> cos(u). Inner: u = x^2 -> 2x. Result: 2x cos(x^2).
Q: What is the slope of the tangent line to y = e^x at x = 0?
A: dy/dx = e^x. At x = 0: e^0 = 1.
Q: If h(x) = f(x)g(x), what is h'(x)?
A: Product rule: h'(x) = f'(x)g(x) + f(x)g'(x).
Q: Find the critical points of f(x) = x^2 - 4x + 5.
A: f'(x) = 2x - 4. Set equal to zero: 2x - 4 = 0, so x = 2.
Q: Find the equation of the tangent line to f(x) = x^2 at the point (3, 9).
A: f'(x) = 2x, so f'(3) = 6. Tangent line: y - 9 = 6(x - 3), which gives y = 6x - 9.
Q: Differentiate y = (x^3 + 1)^10 with respect to x.
A: Chain rule: dy/dx = 10(x^3 + 1)^9 * 3x^2 = 30x^2(x^3 + 1)^9.
Q: Find f'(x) using the quotient rule for f(x) = e^x / x.
A: f'(x) = [e^x * x - e^x * 1] / x^2 = e^x(x - 1) / x^2.
Q: Determine the derivative of f(x) = x * ln(x).
A: Product rule: f'(x) = 1 * ln(x) + x * (1/x) = ln(x) + 1.
Q: Evaluate d/dx [x^e].
A: Power rule with constant exponent e: d/dx [x^e] = e * x^(e-1).
Derivatives feed directly into applications of derivatives (curve sketching, optimisation, related rates) covered in the next set of study notes. The product and quotient rules reappear when you work with more complex integrands. The chain rule is essential for u-substitution in integration. Critical points are the starting point for the first and second derivative tests, which determine local extrema and concavity.
derivative, differentiation, power rule, chain rule, product rule, quotient rule, tangent line, slope, critical point, e^x, ln(x), natural logarithm, exponential function, sin(x), cos(x), composite function, rate of change, instantaneous rate, Calculus I, MSU, Michigan State University, Calculus 1