MAT.126 2.4-2.5

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MAT.126 2.4-2.5

2.5 Implicit Differentiation

Use implicit differentiation to find the derivative of a function

Tangent Line to an Implicit Graph

Higher-Order Implicit Derivatives

Explicit Domains for Implicit Functions

Slope of an Implicit Graph

Graphing Implicit Relations

Process

Chain Rule

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

Distinguish between functions written in implicit form and explicit form

Explicit: y = f(x)

Implicit: y and f(x) are mixed together

2.4 The Chain Rule

Summary of Differentiation Rules

p. 136

Find the derivative of a trigonometric function using the Chain Rule

Tangent Lines

Repeated Application of the Chain Rule

Parentheses

Simplify the derivative of a function using algebra

Powers

Quotients

Factoring Out Least Powers

Find the derivative of a function using the General Power Rule

If y = [u(x)]^n, where u is a differentiable function of x and n is a rational number, then

dy du

--- = n[u(x)]^{n-1} ---

dx dx

or, equivalently

d

---[u^n] = n*u^{n-1} u'

dx

Quotients with Constant Numerators

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

Radicals

Find the derivative of a composite function using the Chain Rule

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 and

dy dy du

--- = --- * ---

dx du dx

or, equivalently

d

---[ f(g(x)) ] = f'(g(x)) g'(x)

dx

Decomposition