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Section 3.5 Implicit Differentiation

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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  1. Section 3.5 Implicit Differentiation

  2. 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΄

  3. Try it If y is some unknown function of x, then

  4. 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.

  5. Example 1 y3 = 2x Solving for y’, we have the derivative

  6. Example 2 x2y3 = -7 Solving for y’, we have

  7. 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 ( ).

  8. Examples Determine dy/dx for the following.

  9. Examples Find the equation of tangent line to the curve

  10. Examples Determine the first derivative of each of the following.

  11. 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′.

  12. Inverse Trig Functions

  13. Examples Find the equation of tangent line to the curve

  14. Group Work

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