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MathematicsA LevelDifferentiation

Quotient rule formula

The quotient rule formula as it is used in A Level Mathematics, taken from the A Level Mathematics formula sheet.

The formula

ddx ⁣(uv)=v u′−u v′v2\dfrac{d}{dx}\!\left(\dfrac{u}{v}\right) = \dfrac{v\,u' - u\,v'}{v^2}

How to use it in the exam

  • Write the formula down before substituting — most mark schemes award a method mark for a correct, clearly quoted formula.
  • Check every quantity is in consistent units before you substitute, then substitute in one step so the working stays easy to follow.
  • Show the rearrangement rather than jumping to the answer, and give the final value to the accuracy the question asks for.

Related differentiation formulas

Power rule
ddx(xn)=nx n−1\dfrac{d}{dx}\big(x^n\big) = n x^{\,n-1}
Product rule
ddx(uv)=udvdx+vdudx\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}
Chain rule
dydx=dydu⋅dudx\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}
Standard derivatives
ddx(sin⁡x)=cos⁡x,ddx(ex)=ex,ddx(ln⁡x)=1x\dfrac{d}{dx}(\sin x) = \cos x, \quad \dfrac{d}{dx}(e^{x}) = e^{x}, \quad \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}

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