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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)=10y9y΄

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

Determine the first derivative of each of the following.

- 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