MH Board · Chapter

Differentiation

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0 / 12

concepts mastered

Composite and inverse functions0/5
Other ways to differentiate0/4
Higher order derivatives0/3

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Up nextThe chain ruleIf $y = f(u)$ and $u = g(x)$ , then $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$ : differentiate the outside, keep the inside, multiply by the inside's derivativeStart

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How the ideas build on each other. Each one opens once you've mastered what it needs.

Composite and inverse functions0 / 5

The chain ruleIf $y = f(u)$ and $u = g(x)$ , then

\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}

:

differentiate the outside, keep the inside, multiply by the inside's derivative

Start

Chain rule with exponentials and logarithms

\dfrac{d}{dx}e^{u} = e^{u}u'

,

\dfrac{d}{dx}a^{u} = a^{u}\log a\cdot u'

,

\dfrac{d}{dx}\log u = \dfrac{u'}{u}

; simplify logs of products firstLocked

Derivative of an inverse functionIf

y = f(x)

has inverse

x = g(y)

, then

g'(y) = \dfrac{1}{f'(x)}

at the matching point: the slopes of a graph and its mirror in

y = x

are reciprocalsLocked

Derivatives of inverse trigonometric functions

\dfrac{d}{dx}\sin^{-1}x = \dfrac{1}{\sqrt{1 - x^2}}

,

\dfrac{d}{dx}\tan^{-1}x = \dfrac{1}{1 + x^2}

,

\dfrac{d}{dx}\sec^{-1}x = \dfrac{1}{|x|\sqrt{x^2 - 1}}

; with the chain rule for

\sin^{-1}u

Locked

Simplifying by substitution firstPut

x = \tan\theta

,

\sin\theta

or

\cos\theta

so that

\tan^{-1}\dfrac{2x}{1 - x^2}

becomes

2\theta = 2\tan^{-1}x

, then differentiate the simple formLocked

Other ways to differentiate0 / 4

Logarithmic differentiationTake $\log$ of both sides to turn products, quotients and powers like $x^{x}$ into sums, differentiate, then multiply back by $y$ Locked
Implicit differentiationDifferentiate every term in $x$ , writing $\dfrac{d}{dx}(y^n) = ny^{n-1}\dfrac{dy}{dx}$ ,
Parametric functionsIf $x = f(t)$ and $y = g(t)$ , then $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ ,
Differentiating one function with respect to anotherThe derivative of $u$ with respect to $v$ is $\dfrac{du}{dv} = \dfrac{du/dx}{dv/dx}$

Higher order derivatives0 / 3

Second and higher derivativesDifferentiate $\dfrac{dy}{dx}$ again for $\dfrac{d^2y}{dx^2}$ ; for $e^{ax}$ , $\sin ax$ and similar, spot the $n$ th derivative's patternLocked
Second derivative of parametric and implicit functionsFor $x = f(t)$ , $y = g(t)$ : $\dfrac{d^2y}{dx^2} = \dfrac{d}{dt}\left(\dfrac{dy}{dx}\right)\Big/\dfrac{dx}{dt}$ ,
Showing a relation between derivativesTo show $(1 - x^2)y_2 - xy_1 + m^2y = 0$ :

then collect the

\dfrac{dy}{dx}

terms and solve

Locked

the slope of the curve traced as

t

varies

Locked

not the ratio of second derivatives

Locked

find

y_1

,

clear roots or fractions, differentiate again, then substitute back

Locked