The Power Rule | CAIE 9709 Pure Mathematics 1 Notes

Pure Mathematics 1

Mathematics9709

Calculate the gradient function directly

Calculating a new chord limit for every function would be slow. The power rule gives the result directly. For a fixed exponent (power) $n$ ,

\frac{d}{dx}(x^n)=nx^{n-1}.

$\dfrac{d}{dx}$ means “differentiate the expression that follows”. Multiply by the exponent, then subtract one from the exponent. For example, $x^5$ gives $5x^{5-1}=5x^4$ . For $x^2$ , it gives the same $2x$ as the chord limit.

MF19 supplies this rule. It applies to the rational powers used in Pure 1 wherever the derivative exists. Roots and negative powers need domain checks, which the next lesson explains.

Keep each coefficient and sign

The coefficient is the number multiplying a term. This constant multiplier scales every change in $y$ , so it scales the gradient too: $3x^2$ differentiates to $3(2x)=6x$ . A constant such as $5$ does not change with $x$ ; its graph is horizontal, so its derivative is zero.

For a sum or difference, add or subtract the derivatives of the individual terms. Thus $x$ gives $1$ , and $-4x$ gives $-4$ , not zero.

Worked example

Find the gradient of $y=2x^3-4x+5$ at $x=2$ .

\begin{aligned}\frac{dy}{dx}&=2(3x^2)-4\\&=6x^2-4.\end{aligned}

Differentiate first, then substitute: the gradient is $6(2)^2-4=20$ . Substituting into the original function instead gives $y=13$ , the point's height.

Find the rule and use it

Your turnIndependent practice

Let $f(x)=3x^4-2x^2+7x-6$ . Find $f'(x)$ and $f'(-1)$ . Does replacing $-6$ by $100$ change either answer?

Show worked answer

f'(x)=12x^3-4x+7.

$f'(-1)=-12+4+7=-1$ . Changing a constant shifts every height by the same amount but does not change any gradient, so neither answer changes.