CAIE A-Level Mathematics 1.7.2 Differentiation Rules

Practise differentiating rational powers, sums and composite expressions with the power and chain rules and simplifying first or second derivatives before substitution.

Exam points

multiply by the original power, reduce the exponent by one and retain constant factors
differentiate a composite bracket by multiplying by the derivative of its inner function
differentiate again or substitute the stated x-value to find the rate of change of gradient

[Maximum number: 4]

The equation of a curve is $y=(3-2 x)^{3}+24 x$ .

Find expressions for $\frac{\mathrm{d} y}{\mathrm{~d} x}$ and $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ .

$\frac{dy}{dx}=3(3-2x)^2(-2)+24=-6(3-2x)^2+24.$
B1, B1

$\frac{d^2y}{dx^2}=-12(3-2x)(-2)=24(3-2x).$
B1FT, B1

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1.7.2—Differentiation rules question 1