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VCE Maths Methods formula sheet

Key formulas for VCE Mathematical Methods: differentiation and integration rules, and probability — a quick study reference.

Differentiation

  • Powerddx(xn)=nxn−1\dfrac{d}{dx}(x^n)=nx^{n-1}
  • Exponentialddx(eax)=aeax\dfrac{d}{dx}(e^{ax})=ae^{ax}
  • Logarithmddx(log⁡ex)=1x\dfrac{d}{dx}(\log_e x)=\dfrac{1}{x}
  • Sineddx(sin⁡ax)=acos⁡ax\dfrac{d}{dx}(\sin ax)=a\cos ax
  • Cosineddx(cos⁡ax)=−asin⁡ax\dfrac{d}{dx}(\cos ax)=-a\sin ax
  • Tangentddx(tan⁡ax)=acos⁡2ax\dfrac{d}{dx}(\tan ax)=\dfrac{a}{\cos^2 ax}
  • Product rule(uv)′=u′v+uv′(uv)'=u'v+uv'
  • Quotient rule(uv)′=u′v−uv′v2\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}
  • Chain ruledydx=dydu⋅dudx\dfrac{dy}{dx}=\dfrac{dy}{du}\cdot\dfrac{du}{dx}

Integration

  • ∫xn dx=xn+1n+1+c, n≠−1\displaystyle\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+c,\ n\neq-1
  • ∫eax dx=1aeax+c\displaystyle\int e^{ax}\,dx=\dfrac{1}{a}e^{ax}+c
  • ∫1x dx=log⁡e∣x∣+c\displaystyle\int \dfrac{1}{x}\,dx=\log_e|x|+c
  • ∫sin⁡ax dx=−1acos⁡ax+c\displaystyle\int \sin ax\,dx=-\dfrac{1}{a}\cos ax+c
  • ∫cos⁡ax dx=1asin⁡ax+c\displaystyle\int \cos ax\,dx=\dfrac{1}{a}\sin ax+c

Probability

  • ConditionalPr⁡(A∣B)=Pr⁡(A∩B)Pr⁡(B)\Pr(A\mid B)=\dfrac{\Pr(A\cap B)}{\Pr(B)}
  • Mean / varianceE(X)=∑x p(x),Var(X)=E(X2)−[E(X)]2E(X)=\sum x\,p(x),\quad \text{Var}(X)=E(X^2)-[E(X)]^2
  • BinomialPr⁡(X=x)=(nx)px(1−p)n−x\Pr(X=x)=\binom{n}{x}p^x(1-p)^{n-x}
  • Binomial mean/varμ=np,σ2=np(1−p)\mu=np,\quad \sigma^2=np(1-p)
  • Standardise (normal)Z=X−μσZ=\dfrac{X-\mu}{\sigma}

A study reference compiled by ATAR99. The official VCAA formula/data sheet is provided in your exam — always use that as the authority on the day.

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