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Section 3.5 Implicit Differentiation PowerPoint PPT Presentation


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Section 3.5 Implicit Differentiation. Example. If f ( x ) = ( x 7 + 3 x 5 – 2 x 2 ) 10 , determine f ’ (x) . Answer: f ΄ ( x ) = 10( x 7 + 3 x 5 – 2 x 2 ) 9  (7 x 6 + 15 x 4 – 4 x ). Now write the answer above only in terms of y if y = ( x 7 + 3 x 5 – 2 x 2 ) .

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Section 3.5 Implicit Differentiation

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Section 3.5 Implicit Differentiation


Example

If f(x) = (x7 + 3x5 – 2x2)10, determine f ’(x).

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)=10y9y΄


Try it

If y is some unknown function of x, then


Purpose

9x + x2– 2y = 5

5x – 3xy + y2 = 2y

Easy to solve for y and differentiate

Not easy to solve for y and differentiate

In equations like 5x – 3xy + y2 = 2y, we simply assume that y = f(x), or some function of x which 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 terms containing y, we use the chain rule.


Example 1

y3 = 2x

Solving for y’, we have the derivative


Example 2

x2y3 = -7

Solving for y’, we have


Implicit Differentiation

  • Differentiate both sides of the equation:

    Since y is a function of x, every time we differentiate a term containing y, we need to multiply it by y’or dy/dx

  • Solve for y’:

    • Every term containing y’ should be moved to the left by adding or subtracting terms only.

    • Every term containing no y’ should be moved to the right hand side.

    • Factor out y’ and divide both sides by the expression inside ( ).


Examples

Determine dy/dx for the following.


Examples

Find the equation of tangent line to the curve


Examples

Determine the first derivative of each of the following.


Logarithmic Differentiation

  • Take the natural logarithms of both sides of an equation y = f(x).

  • Use the laws of logarithms to expand the expression.

  • Differentiate implicitly with respect to x.

  • Solve the resulting equation for y′.


Inverse Trig Functions


Examples

Find the equation of tangent line to the curve


Group Work


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