MAT.126
2.4-2.5

2.4 The Chain Rule

Find the derivative of a composite function using the Chain Rule

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If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x anddy dy du--- = --- * ---dx du dxor, equivalently d---[ f(g(x)) ] = f'(g(x)) g'(x)dx

Decomposition

Find the derivative of a function using the General Power Rule

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If y = [u(x)]^n, where u is a differentiable function of x and n is a rational number, thendy du--- = n[u(x)]^{n-1} ---dx dxor, equivalently d---[u^n] = n*u^{n-1} u'dx

Radicals

Quotients with Constant Numerators

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d---[ sin u ] = (cos u) u'dx d---[ cos u ] = -(sin u) u'dx d---[ tan u ] = (sec^2 u) u'dx d---[ cot u ] = -(csc^2 u) u'dx d---[ sec u ] = (sec u tan u) u'dx d---[ csc u ] = -(csc u cot u) u'dx

Simplify the derivative of a function using algebra

Factoring Out Least Powers

Quotients

Powers

Find the derivative of a trigonometric function using the Chain Rule

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d---[ sin u ] = (cos u) u'dx d---[ cos u ] = -(sin u) u'dx d---[ tan u ] = (sec^2 u) u'dx d---[ cot u ] = -(csc^2 u) u'dx d---[ sec u ] = (sec u tan u) u'dx d---[ csc u ] = -(csc u cot u) u'dx

Parentheses

Repeated Application of the Chain Rule

Tangent Lines

Summary of Differentiation Rules

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p. 136

2.5 Implicit Differentiation

Distinguish between functions written in implicit form and explicit form

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Explicit: y = f(x)Implicit: y and f(x) are mixed together

Use implicit differentiation to find the derivative of a function

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Differentiate both sides of the equation with respect to x.Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.Factor dy/dx out of the left side of the equation.Solve for dy/dx.

Chain Rule

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To differentiate an implicitly defined function one must use the chain rule on all terms involving y. d dy---[ f(y) ] = f'(y) ----dx dx

Process

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Differentiate both sides of the equation with respect to x.Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.Factor dy/dx out of the left side of the equation.Solve for dy/dx.

Graphing Implicit Relations

Slope of an Implicit Graph

Explicit Domains for Implicit Functions

Higher-Order Implicit Derivatives

Tangent Line to an Implicit Graph