Derivatives

Learn how to compute the average rate of change of a function over an interval, interpret it as the slope of a secant line, and see why the average depends on the interval chosen.

Find how fast a function is changing at a single instant by watching average rates over shrinking intervals settle on one value, and read that instantaneous rate of change as the slope of the tangent line.

Compute the derivative $f'(a)$ from its limit definition - forming and simplifying the difference quotient, then taking the limit - and use it to write the equation of the tangent line at a point.

Extend the derivative from a single point to a function: find $f'(x)$ from the limit definition for polynomials and evaluate it to get the slope at any point, then sketch the graph of $f'$ from the graph of $f$.

Recognise where a function fails to be differentiable - at corners, cusps, vertical tangents, and discontinuities - and test differentiability at a point by comparing the left-hand and right-hand derivatives.

Learn how to read, write and translate between the different notations for the derivative, and read off its value at a single point.

Learn the first two rules for differentiating by formula: the derivative of any constant is zero, and the power rule for differentiating any power of $x$. Rewrite roots and reciprocals in exponent form first so the power rule applies.

Learn to differentiate constant multiples, sums and differences: a constant factor passes straight through differentiation, and a sum or difference differentiates term by term. Combine these with the power rule to differentiate any polynomial.

Learn to differentiate a product of two functions with the product rule, and why the derivative of a product is not the product of their derivatives.

Differentiate quotients of functions using the quotient rule, and recognise when a quotient simplifies so the rule is not needed.

Break a composite function into its inner and outer parts, and find an overall rate of change by multiplying the rates of each linked stage.

Differentiate composite functions - powers, roots and reciprocals of polynomials - with the chain rule, and choose the quickest method for a mix of functions.

Differentiate the natural exponential function $e^x$ and composite functions $e^{g(x)}$.

Differentiate exponential functions of any base, $a^x$, and composite functions $a^{g(x)}$.

Differentiate the natural logarithm $\ln x$ and composite functions $\ln g(x)$, then extend the rule to $\ln \lvert x \rvert$, which is defined on both sides of zero.

Differentiate logarithms of any base, $\log_a x$, and composite functions $\log_a g(x)$.

Differentiate $\sin x$ and $\cos x$, and composite functions $\sin(g(x))$ and $\cos(g(x))$.

Differentiate $\tan x$ and composite functions $\tan(g(x))$.

Recognise when a composite function needs the chain rule more than once, and differentiate it by applying the rule once per layer.

Differentiate products and quotients in which one part is a composite function, using the chain rule alongside the product and quotient rules.

Compute the second derivative $f''$ and higher-order derivatives $f^{(n)}$, and use a polynomial's degree to predict the order at which its derivatives vanish.

State a high-order derivative of an exponential, a sine or a cosine from the repeating pattern its derivatives follow, instead of differentiating many times.

Find every critical point of a function - the points where $f'(x) = 0$ or $f'$ has no value - and read from a graph whether each one is a local maximum, a local minimum, or neither.

Use the sign of the first derivative $f'$ to find where a function is increasing or decreasing, and to classify each critical point as a local maximum, a local minimum, or neither.

Find where a curve is concave up or concave down using the sign of the second derivative, and locate the inflection points where its concavity changes.

Classify a critical point from the sign of $f''$, and choose the right test - using the first derivative test when the second is inconclusive ($f''(c)=0$) or does not apply ($f'(c)$ undefined).

Learn how to find the largest and smallest values a function reaches on a closed interval are its global extrema. Find them with the closed-interval method: evaluate the function at every critical point and endpoint, then compare.

Learn how to turn a described situation into the two equations an optimisation problem needs: an objective function for the quantity to be made largest or smallest, and a constraint equation for what the situation holds fixed.

Learn how to use the constraint equation to eliminate one unknown, turning a two-variable objective function into a function of a single variable. Then find its domain: the values the remaining variable is allowed to take.

Learn how to find the optimum of an objective function by solving $f'(x) = 0$ and discarding any critical point its domain does not allow. Then certify that optimum as global by comparing its value against the two ends of the domain.

Work an optimisation problem end to end from a written description: build the objective function and the constraint, reduce to one variable, find the domain the situation allows, and confirm the optimum is global before reporting the answer.