Differentiation

Multiply by the power, then knock the power down by one:

It works for every rational power — negative and fractional included — which is why P1 questions arrive dressed in roots and fractions: rewrite first, then differentiate.

Constants vanish; sums differentiate term by term. Most P1 slips on this rule happen in the rewriting, not the calculus.

For a function of a function, differentiate the outside, keep the inside, multiply by the inside’s derivative:

In P1 the chain rule appears wherever a bracket carries a power — including negative powers like $y=(3x+1{)}^{-2}$ — and in connected rates of change (below). The factor from the inside (here $4x$) is the mark examiners look for.

The derivative evaluated at a point is the gradient of the tangent there. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal:

1. Differentiate.
2. Substitute the $x$-coordinate to get a numerical gradient.
3. Pick the right gradient for what was asked — tangent uses $m$, normal uses $-1\mathrm{/}m$.
4. Line through the point: $y-y_1=m(x-x_1)$.

Stationary points solve $\frac{dy}{dx}=0$. Their nature comes from the second derivative:

A function is increasing where $\frac{dy}{dx}>0$ and decreasing where $\frac{dy}{dx}<0$ — P1 asks this as “find the set of values of $x$ for which $f$ is decreasing”, which is a quadratic inequality in disguise.

When two quantities both change with time, the chain rule links their rates:

A typical P1 question: the radius of a circle increases at 0.2 cm s⁻¹; find how fast the area grows when $r=5$. From $A=\pi r^2$, $\frac{dA}{dr}=2\pi r$, so $\frac{dA}{dt}=2\pi (5)\times 0.2=2\pi$ cm² s⁻¹. Write the chain-rule statement first — it is usually its own mark.

1. 1.Giving the tangent when the normal was asked (and vice versa)

   Cambridge examiner reports name this mix-up every session. Underline the word in the question, and after computing $m$, write one explicit line: $m_{\text{normal}}=-1\mathrm{/}m$.

2. 2.Forgetting the chain-rule factor

   $(2x^2-5{)}^6$ does not differentiate to $6(2x^2-5{)}^5$ — the $\times 4x$ is where the mark lives. If the bracket’s inside isn’t just $x$, a factor is owed.

3. 3.Not rewriting roots and fractions before differentiating

   Convert $\sqrt{x}$ to $x^{1\mathrm{/}2}$ and $\frac{k}{x^n}$ to $k{x}^{-n}$ first. Attempting the power rule on $2\mathrm{/}{x}^2$ by sight produces sign errors almost every time.

4. 4.Solving dy/dx = 0 on the calculator with no working

   9709’s standing rule: unsupported answers from a calculator earn no marks. Show the factorisation or formula line before stating roots.

5. 5.Testing nature with y-values instead of the second derivative

   Substituting the stationary $x$ into $y$ tells you the height, not the nature. Use $\frac{d^{2}y}{d{x}^2}$ (or a signed gradient table) and state the conclusion with its evidence.