Warm Up – NO CALCULATOR

1) Find the derivative of y = x2 ln(x3)

2) A particle moves along the x-axis so that at any time t > 0 its velocity is given by v(t) = tlnt – t.

a) Write an expression for the acceleration of the particle.

b) For what values of t is the particle moving to the right?

Derivatives of logDerivatives of logbb

and and Logarithmic Logarithmic

DifferentiationDifferentiation

Derivatives of logDerivatives of logbb

and and Logarithmic Logarithmic

DifferentiationDifferentiation

To determine the derivatives of logb (not ln),

use the change of base formula to change it to ln.

Examples

Find the derivative.

8log ( )y x

3log 4 1y x

2log

4

xy

x

All it means is that you have something you want to

differentiate, you take the natural log of both sides of an equation before you take the derivative.

Logarithmic Differentiation…

Why would you want to do that?

Reason #1Reason #1

Take the derivative of 2 3 7 2( 3)( 4)( 5)(3 13)y x x x x

Let’s Review the steps…

1. Take ln of both sides.2. Simplify the right side using log

properties.3. Take the derivative of both sides. (You always get on the left side)

4. Move the y to the right side of the = (to solve for dy/dx) and substitute the original y equation in for y.

1 dy

y dx

Your Turn Use logarithmic differentiation to

determine the derivative of

4

5

(3 1)( 2)

(2 5)( 6)

x xy

x x

Why would you want to do that?

Reason #2Whenever you have an x in the exponent

3xy xy x

You again!Find the derivative of

23xy x