3'5 Implicit Differentiation

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3.5 Implicit Differentiation
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Try It
If f(x) (x7 3x5 2x2)10, determine y?.
Answer f?(x) 10(x7 3x5 2x2)9?(7x6 15x4
4x)
Now write the answer above only in terms of y if
y (x7 3x5 2x2).
Answer f ?(x) dy10/dx 10y9?y?
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Try It
If y is some unknown function of x, determine
dy35 / dx.
Answer dy35 / dx 35y34?y?
In general, if y is some function of x, then what
is dyn / dx?
Answer dyn / dx ny (n 1)?y?
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Power Rule for Implicit Differentiation
If y is some unknown function of x, then
using the chain rule.
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Purpose
Which of these would be easy to solve for y and
differentiate?
9x x2 2y 5
5x 3xy y2 2y
Easy to solve for y and differentiate
Not easy to solve for y and differentiate
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The Idea
In equations like 5x 3xy y2 2y, we simply
assume that y f(x), or some function of x which
we is not easy to find.
Process wise, simply take the derivative of each
side of the equation with respect to x and when
we encounter a yn, we use
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The Basics Example 1
For example, y3 2x
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The Basics Example 1
So,
Solving for dy/dx, we have the derivative
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The Basics Example 2
If there is a product, we will need the product
rule. For example, x2y3 -7
Must Use The Product Rule
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The Basics Example 1
So,
Solving for dy/dx, we have the derivative
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You Try It
Determine dy/dx for the following.
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Derivative of Inverse Trig Functions
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Derivative of Inverse Trig Functions
All the formulas for the inverse trigonometric
functions are on back flap of your text.
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Application Family of Functions
The function f(x) kx2, where k can be any real
number defines a family of functions. The graphs
of such a family is called a family of curves. If
we let k be a specific value, we call the
function a particular function and the graph the
particular curve.
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Application Orthogonal Trajectories
Two curves are orthogonal if at each point of
intersection their tangent lines are
perpendicular. If a family of functions are
orthogonal, the curves are said to be orthogonal
trajectories of each other.
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Application Orthogonal Trajectories
Show that the given families of curves are
orthogonal trajectories of each other. 1. y
ax3 and x2 3y2 b 2. x2 y2 ax and x2
y2 by