7 Differentiation — notes on one page

7.1 · Derivative of f(x) as tangent gradient and as a limit; rate of change; sketch the gradient function; first-principles differentiation for small integer powers of x, sin x, cos x; second derivatives; convexity, concavity, inflection.

• The derivative $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ is the tangent gradient and instantaneous rate of change. For $\sin x$ and $\cos x$, expand the compound angle and use $\lim_{h\to0}\frac{\sin h}{h}=1$ and $\lim_{h\to0}\frac{\cos h-1}{h}=0$.
• To sketch $f'$, record where $f$ rises or falls, where its tangents are horizontal, and how steep those tangents are; the zeros of $f'$ occur at stationary points of $f$. The second derivative is the rate of change of the gradient: $f''(x)>0$ indicates a convex section and $f''(x)<0$ a concave section.
• At an inflection point, $f''$ changes sign.
• A common error is to declare an inflection point from $f''(x)=0$ alone; a sign change of concavity must be checked.
• For first principles, use $f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$; expanding $(x+h)^2$ or $(x+h)^3$ and cancelling before taking the limit derives the required power result.

Exam tip. For first principles, state the limiting process explicitly at least once before stating the derivative — $h\to0$ or $\delta x\to0$, whichever letter you used. On a gradient-function sketch, use zeros for stationary points and whether the gradient is rising or falling for convexity.

7.2 · Differentiate xⁿ for rational n, and related sums, differences and constant multiples; differentiate e^(kx), a^(kx), sin kx, cos kx, tan kx; understand and use the derivative of ln x.

Definition. Specification 7.2: differentiate $x^n$ for rational values of $n$, and related constant multiples, sums and differences; differentiate $e^{kx}$ and $a^{kx}$, $\sin kx$, $\cos kx$, $\tan kx$ and related sums, differences and constant multiples; understand and use the derivative of $\ln x$.

• Multiply by the power and reduce it by one: $\dfrac{d}{dx}(x^n)=nx^{n-1}$ for any rational $n$, so rewrite roots and reciprocals as powers first, $\sqrt x=x^{1/2}$ and $\dfrac{3}{x^2}=3x^{-2}$, and a fraction of powers is split term by term before differentiating.
• Each term is differentiated separately and a constant disappears; there is no $+c$ in a derivative.
• The standard results are $\dfrac{d}{dx}(e^{kx})=ke^{kx}$, $\dfrac{d}{dx}(a^{kx})=k\ln a\,a^{kx}$, $\dfrac{d}{dx}(\sin kx)=k\cos kx$, $\dfrac{d}{dx}(\cos kx)=-k\sin kx$, $\dfrac{d}{dx}(\tan kx)=k\sec^2kx$, with $x$ in radians, and $\dfrac{d}{dx}(\ln x)=\dfrac1x$; the factor $k$ from the inside is the usual omission.
• A second derivative is the derivative of the first.
• An exponential model $m=Ae^{kt}$ satisfies $\dfrac{dm}{dt}=km$, so the rate is proportional to the quantity.

What the examiner wants. Clean derivatives with no stray constant. The June 2024 Paper 2 report says the commonest way to lose a mark on the opening question was a spurious “$+c$” attached to a derivative, with a few dividing by the index instead of multiplying by it, and some “simplifying” the second derivative $24x-14$ to $12x-7$.

Exam tip. Differentiate each term separately and display the chain-rule multiplier for every composite term.

7.3 · Apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, points of inflection; identify where functions are increasing or decreasing.

Definition. Specification 7.3: apply differentiation to find gradients, tangents and normals, maxima and minima and stationary points, and points of inflection; identify where functions are increasing or decreasing.

• The gradient at $x=a$ is $f'(a)$; the tangent there is $y-f(a)=f'(a)(x-a)$ and the normal has gradient $-1/f'(a)$.
• Stationary points solve $f'(x)=0$, and their nature comes from the second derivative, $f''>0$ a minimum and $f''<0$ a maximum, or from the sign of $f'$ either side; a stated calculation and a reason are needed, not just the word.
• A point of inflection needs $f''(x)=0$ and a change of sign in $f''$, so evaluate $f''$ on both sides.
• A function is increasing where $f'(x)>0$ and decreasing where $f'(x)<0$: when $f'$ is a quadratic, find its critical values algebraically, sketch or test the sign, and give the answer as inequalities in $x$.
• Optimisation questions differentiate a formula in one variable, set the derivative to zero, then justify the maximum or minimum and answer in context with units.

What the examiner wants. Algebra when the question says “using algebra”. The June 2024 Paper 1 report on the decreasing-function question says not all candidates observed that demand, and those who solved the quadratic on a calculator or gave an incorrect factorisation forfeited the first mark, even with the right critical values.

Exam tip. For stationary-point questions, solve the derivative equation, classify each point and finish with a derivative sign chart.

7.4 · Differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.

Definition. Specification 7.4: differentiate using the product rule, the quotient rule and the chain rule, including problems involving connected rates of change and inverse functions.

• Chain rule: differentiate the outer function, keep the inside unchanged, multiply by the derivative of the inside, so $e^{4x^2-1}$ gives $8xe^{4x^2-1}$ and $(2x+1)^5$ gives $10(2x+1)^4$; writing $u=4x^2-1$ as an intermediate step is what the reports recommend. Product rule: $(uv)'=u'v+uv'$.
• Quotient rule: $\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}$, with the order of the numerator fixed and brackets kept round each product. After differentiating, factorise out the common factor, $(2x+1)^3(10x+1)$, because the factorised form is usually what the next part needs.
• Connected rates use $\dfrac{dV}{dt}=\dfrac{dV}{dh}\times\dfrac{dh}{dt}$: differentiate the formula first, then substitute the numbers.
• For an inverse function or $x$ given in terms of $y$, $\dfrac{dy}{dx}=1\Big/\dfrac{dx}{dy}$.
• The derivatives of $\sec$, $\cosec$ and $\cot$ are in the formulae booklet.

What the examiner wants. An intermediate step for the chain rule, then factorise rather than expand. The June 2024 Paper 2 report says candidates struggled with the exponential $e^{4x^2-1}$ and the most successful wrote a substitution step before applying the chain rule, and that expanding brackets instead of using the chain rule and factorisation made a later question longer and less accurate.

Exam tip. Name the outermost rule first, differentiate nested functions carefully, then simplify only after every factor is present.

7.5 · Differentiate simple functions and relations defined implicitly or parametrically, for first derivative only.

• For an implicit relation, differentiate every term with respect to $x$ and attach a factor $\frac{\mathrm dy}{\mathrm dx}$ whenever a differentiated term contains $y$.
• For parametric equations $x=x(t)$ and $y=y(t)$, use $\frac{\mathrm dy}{\mathrm dx}=\frac{\mathrm dy/\mathrm dt}{\mathrm dx/\mathrm dt}$ where $\mathrm dx/\mathrm dt\ne0$.
• After finding the gradient, use the parameter or relation to obtain the actual point before writing a tangent or normal equation.
• A common error in implicit differentiation is to write $\frac{\mathrm d}{\mathrm dx}(xy)=y+x$ instead of $y+x\frac{\mathrm dy}{\mathrm dx}$.

Exam tip. For a parametric tangent, evaluate both parameter derivatives at the stated parameter before forming the gradient and line equation.

7.6 · Construct simple differential equations in pure mathematics and in context (contexts may include kinematics, population growth and modelling the relationship between price and demand).

• Translate a rate statement into derivative notation after defining the dependent and independent variables, including their units where relevant.
• Phrases such as 'proportional to' introduce a positive constant $k$; words such as 'decreases' or 'decays' determine whether a minus sign is needed.
• A limiting or equilibrium value often appears as a difference, for example growth towards a capacity $K$ can be modelled by $\frac{\mathrm dP}{\mathrm dt}=k(K-P)$.
• A common error is to solve the differential equation when only its construction is requested, while failing to state or determine the proportionality constant.

Exam tip. Translate each verbal rate statement into a signed differential equation, then substitute the calibration data to find the constant.