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Differentiation rules
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Derivatives ofΒ βxr,ex,lnx,sinxβΒ andΒ βcosx.Β
Derivative of xβΏ where n is an integer
SL 5.3
β
f(x)=xn,Β nβZβfβ²(x)=nxnβ1π
βDerivatives of sums and scalar multiples
SL 5.3
β
dxdβ(af(x))=afβ²(x)π«
ββ
dxdβ(f(x)+g(x))=fβ²(x)+gβ²(x)π«
ββ
dxdβ(af(x)+bg(x))=afβ²(x)+bgβ²(x)π«
βDerivative of xβΏ where n is rational
AHL AI 5.9
β
f(x)=xn,Β nβQβfβ²(x)=nxnβ1π
βDerivative of e^x
AHL AI 5.9
β
f(x)=exβfβ²(x)=exπ
βDerivative of ln
AHL AI 5.9
β
f(x)=lnxβfβ²(x)=x1βπ
βDerivatives of sin and cos
AHL AI 5.9
β
f(x)=sinxβfβ²(x)=cosxπ
ββ
g(x)=cosxβgβ²(x)=βsinxπ
βChain rule
AHL AI 5.9
The chain rule tells you how to find the derivative of a function that is composed with another function. It can be expressed a few different ways:
β
(g(f(x)))β²=gβ²(f(x))β
fβ²(x)π«
ββ
y=g(u) where u=f(x)
ββ
dxdyβ=dudgββ
dxduβπ
βIntuitively, what the chain rule is essentially saying is that the rate of change of the composite function is the rate of change of the outside function when you change the inside function times the rate of change of the inside function when you changeΒ βx.Β
Product and Quotient rule
AHL AI 5.9
The product and quotient rules are given by
β
(uv)β²=uβ²v+vβ²uπ
ββ
(vuβ)β²=v2uβ²vβvβ²uβπ
βDerivative of tan(x)
AHL AI 5.9
β
f(x)=tanxβfβ²(x)=sec2(x)π
βNice work completing Differentiation rules, here's a quick recap of what we covered:
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