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Differentiation rules
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Derivatives of xr,ex,lnx,sinx and cosx.
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
SL 5.6
f(x)=xn, n∈Q⇒f′(x)=nxn−1📖
Derivative of e^x
SL 5.6
f(x)=ex⇒f′(x)=ex📖
Derivative of ln
SL 5.6
f(x)=lnx⇒f′(x)=x1📖
Derivatives of sin and cos
SL 5.6
f(x)=sinx⇒f′(x)=cosx📖
g(x)=cosx⇒g′(x)=−sinx📖
Chain rule
SL 5.6
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
SL 5.6
The product and quotient rules are given by
(uv)′=u′v+v′u📖
(vu)′=v2u′v−v′u📖
Derivative of tan(x)
AHL 5.15
f(x)=tanx⇒f′(x)=sec2(x)📖
Derivatives of reciprocal trig functions
AHL 5.15
f(x)=secx⇒f′(x)=secxtanx📖
g(x)=cosecx⇒g′(x)=−cosecxcotx📖
h(x)=cotx⇒h′(x)=−cosec2(x)📖
Derivatives of aˣ and logₐ(x)
AHL 5.15
f(x)=ax⇒f′(x)=axlna📖
g(x)=logax⇒f′(x)=xlna1📖
Derivatives of arcsin, arccos and arctan
AHL 5.15
f(x)=arcsinx⇒f′(x)=√1−x21📖
f(x)=arccosx⇒f′(x)=−√1−x21📖
f(x)=arctanx⇒f′(x)=1+x21📖
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