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2.3 Differentiation

Syllabus
9231–2028–2029
Topic
2.3
Level
A2

Learning objectives

2.3.1Hyperbolic functions• differentiate hyperbolic functions and differentiate sin-1x, cos-1x, sinh-1x, cosh-1x and tanh-1x2.3.2An expression for x y d• obtain an expression for x y d d 2 2 in cases where the relation between x and y is defined implicitly or parametrically2.3.3Maclaurin series• derive and use the first few terms of a Maclaurin's series for a function. Derivation of a general term is not included, but successive 'implicit' differentiation steps may be required, e.g. for y = tan x following an initial differentiation rearranged as y′= 1 + y2.

Differentiate hyperbolic and inverse functions with the right sign and domain

f(x)f(x) f′(x)f'(x)
sinh⁡x\sinh x cosh⁡x\cosh x
cosh⁡x\cosh x sinh⁡x\sinh x
tanh⁡x\tanh x sech⁡2x\operatorname{sech}^2x
sech⁡x\operatorname{sech}x −sech⁡xtanh⁡x-\operatorname{sech}x\tanh x
cosech⁡x\operatorname{cosech}x −cosech⁡xcoth⁡x-\operatorname{cosech}x\coth x
coth⁡x\coth x −cosech⁡2x-\operatorname{cosech}^2x
f(x)f(x) f′(x)f'(x) real-domain condition
sin⁡−1x\sin^{-1}x 1/1−x21/\sqrt{1-x^2} −1<x<1-1<x<1
cos⁡−1x\cos^{-1}x −1/1−x2-1/\sqrt{1-x^2} −1<x<1-1<x<1
sinh⁡−1x\sinh^{-1}x 1/1+x21/\sqrt{1+x^2} all real xx
cosh⁡−1x\cosh^{-1}x 1/x2−11/\sqrt{x^2-1} x>1x>1
tanh⁡−1x\tanh^{-1}x 1/(1−x2)1/(1-x^2) ∣x∣<1|x|<1

Foracomposite,multiplybytheinnerderivative.Forexample,For a composite, multiply by the inner derivative. For example,\frac{d}{dx}\left[\cosh^{-1}(2x)\right]=\frac{2}{\sqrt{(2x)^2-1}}=\frac{2}{\sqrt{4x^2-1}},\qquad x>\tfrac12.

Here f−1f^{-1} means an inverse function, not 1/f1/f. The derivative of cosh⁡x\cosh x is positive sinh⁡x\sinh x; the negative sign belongs to sech⁡x\operatorname{sech}x, cosech⁡x\operatorname{cosech}x and coth⁡x\coth x.

Differentiate again without first making y explicit

For an implicit relation F(x,y)=0F(x,y)=0, differentiate every term with respect to xx, treating yy as y(x)y(x), and solve for y′y'. Differentiate that whole equation again: product and chain rules now produce both y′y' and y′′y''. Substitute the known point and first derivative only after the second differentiated equation is complete.

If $x^2+xy+y^2=7$, then2x+y+(x+2y)y'=0.At $(1,2)$, $y'=-4/5$. Differentiating again gives2+2y'+2(y')^2+(x+2y)y''=0,sosoy''=-\frac{2+2(-4/5)+2(16/25)}{5}=-\frac{42}{125}.

When $x=x(t)$ and $y=y(t)$ with $dx/dt\ne0$,\frac{dy}{dx}=\frac{dy/dt}{dx/dt},\qquad \frac{d^2y}{dx^2}=\frac{1}{dx/dt}\frac{d}{dt}\left(\frac{dy}{dx}\right)=\frac{x'y''-y'x''}{(x')^3},where primes in the final fraction mean derivatives with respect to $t$.

For $x=t^2+1$ and $y=t^3$,\frac{dy}{dx}=\frac{3t^2}{2t}=\frac32t,hencehence\frac{d^2y}{dx^2}=\frac{1}{2t}\frac{d}{dt}\left(\frac32t\right)=\frac{3}{4t}.At $t=1$, the second derivative is $3/4$.

For a parametric curve, d/dt(dy/dx)d/dt(dy/dx) is not yet d2y/dx2d^2y/dx^2; divide by dx/dtdx/dt once more. If dx/dt=0dx/dt=0, this formula cannot be used at that parameter value without separate analysis.

Build the first Maclaurin terms from derivatives at zero

ThefirsttermsoftheMaclaurinseriesareThe first terms of the Maclaurin series aref(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots.To derive terms through $x^n$, find only the derivatives through order $n$, evaluate each at zero, and divide by its factorial. A general term is not required here.

For $y=\tan x$, start from $y'=1+y^2$. At $x=0$, $y=0$ and $y'=1$. Theny''=2yy'\Rightarrow y''(0)=0,y'''=2(y')^2+2yy''\Rightarrow y'''(0)=2.ThereforeTherefore\tan x=x+\frac{x^3}{3}+\cdots.Thisissuccessiveimplicitdifferentiation:noformulaforthegeneralderivativewasneeded.This is successive implicit differentiation: no formula for the general derivative was needed.

A truncated series can simplify a nearby calculation. From $e^{-x^2}=1-x^2+\cdots$,\int_0^{1/5}e^{-x^2},dx\approx\int_0^{1/5}(1-x^2),dx=\left[x-\frac{x^3}{3}\right]_0^{1/5}=\frac{74}{375}.

Do not omit the factorials or keep terms beyond the requested power. If a known series is substituted into another expression, retain enough source terms to produce every requested final power before simplifying.