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Differentiation rules
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Derivatives of xr,ex,lnx,sinx and cosx. The chain, product and quotient rules, as well as the derivatives of of tanx.
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Derivative of xⁿ where n is an integer
SL 5.3
f(x)=xn, n∈Z⇒f′(x)=nxn−1📖
Derivatives of sums and scalar multiples
SL 5.3
dxd(af(x))=af′(x)🚫
dxd(f(x)+g(x))=f′(x)+g′(x)🚫
dxd(af(x)+bg(x))=af′(x)+bg′(x)🚫
Derivative of xⁿ where n is rational
AHL AI 5.9
f(x)=xn, n∈Q⇒f′(x)=nxn−1📖
Derivative of e^x
AHL AI 5.9
f(x)=ex⇒f′(x)=ex📖
Derivative of ln
AHL AI 5.9
f(x)=lnx⇒f′(x)=x1📖
Derivatives of sin and cos
AHL AI 5.9
f(x)=sinx⇒f′(x)=cosx📖
g(x)=cosx⇒g′(x)=−sinx📖
Chain 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:
(g(f(x)))′=g′(f(x))⋅f′(x)🚫
y=g(u) where u=f(x)
dxdy=dudg⋅dxdu📖
Intuitively, 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 change x.
Product and Quotient rule
AHL AI 5.9
The product and quotient rules are given by
(uv)′=u′v+v′u📖
(vu)′=v2u′v−v′u📖
Derivative of tan(x)
AHL AI 5.9
f(x)=tanx⇒f′(x)=sec2(x)📖
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