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Differentiation rules
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Power rule, chain rule, product rule, quotient rule, derivatives of ex, ax, loga(x),
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📖
Chain rule
SL 5.6
(g(f(x)))′=g′(f(x))⋅f′(x)🚫
y=g(u) where u=f(x)
dxdy=dudg⋅dxdu📖
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📖
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📖
Derivatives of sin and cos
SL 5.6
f(x)=sinx⇒f′(x)=cosx📖
g(x)=cosx⇒g′(x)=−sinx📖
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📖
Implicit Differentiation
AHL 5.14
Implicit differentiation is when we differentiate both sides of an equation. It is helpful when we have an equation that cannot be simplified to y=f(x).
Implicit Derivative: Horizontal and vertical tangents & normals
In problems involving implicit derivatives, you may be asked to solve for points where the tangent to the curve is horizontal or vertical. A horizontal tangent means dxdy=0, and a vertical tangent occurs in the case where dxdy=denominatornumerator and the denominator equals zero.
If the question asks for vertical / horizontal normals, just recall that a vertical normal means a horizontal tangent, and vice-versa.
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