Calculus essentials

Rearrange without changing the equation

Apply the same operation to both sides. For \(3(y-2)=x\), divide by 3, then add 2: \(y=x/3+2\). Before cancelling a factor, state when it is nonzero. Cancelling \(x\) from \(x(x-1)=0\) would lose \(x=0\).

Use logarithm laws with their domains

For positive \(a,b\), \(\ln(ab)=\ln a+\ln b\) and \(\ln(a/b)=\ln a-\ln b\). A real logarithm needs a positive argument. In integration, \(\int 1/x\,dx=\ln|x|+C\), on an interval avoiding zero.

Choose a differentiation rule

Simplify first when helpful: \((x^2+1)/x=x+1/x\), for \(x\ne0\). A composition needs the chain rule; a product needs \((uv)'=u'v+uv'\); a quotient needs \((u/v)'=(u'v-uv')/v^2\). For \((3x+1)^2\), include the inner derivative 3.

Use radians in calculus

The formulas \((\sin x)'=\cos x\) and \((\tan x)'=\sec^2x\) assume radians. Convert degrees with \(\theta_{\rm rad}=\theta_{\rm deg}\pi/180\). Check the calculator mode when evaluating a derivative at \(\pi/3\).

Read a derivative’s sign

On an interval, \(f'>0\) means increasing and \(f'\lt0\) means decreasing. It does not tell you whether \(f\) itself is positive. For classification, check signs on both sides of a stationary point; a zero derivative alone does not prove a maximum.

Keep the integration constant

An indefinite integral represents a family: \(\int 2x\,dx=x^2+C\). If the curve passes through \((1,3)\), then \(C=2\). Use a fresh constant when integrating a second time. Differentiate your result to check the integrand.

Check restrictions and branches

A denominator cannot be zero. Real square roots need a nonnegative radicand, and \(\sqrt{u^2}=|u|\). Squaring may add roots, so substitute back. A differential-equation solution must stay on an interval where the original equation is defined; an algebraic formula on the other side of a singularity is not automatically the same initial-value solution.

What needs to be shown?

Show the derivatives, antiderivatives or equations the question asks for, and connect the steps that lead to your conclusion. A final number alone may leave the required reasoning unshown. In an optimisation problem, state your variable and constraint, find an admissible candidate, and justify its nature unless the question says you may assume it. Keep exact values until rounding is needed.

Check the official 2026 Level 3 Calculus assessment specifications for current requirements.