warm up – no calculator 1)find the derivative of y = x 2 ln(x 3 ) 2)a particle moves along the...

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Warm Up – NO CALCULATOR 1) Find the derivative of y = x 2 ln(x 3 ) 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?

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Page 1: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

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?

Page 2: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

Derivatives of logDerivatives of logbb

and and Logarithmic Logarithmic

DifferentiationDifferentiation

Derivatives of logDerivatives of logbb

and and Logarithmic Logarithmic

DifferentiationDifferentiation

Page 3: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

To determine the derivatives of logb (not ln),

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

Page 4: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

Examples

Find the derivative.

8log ( )y x

3log 4 1y x

2log

4

xy

x

Page 5: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

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…

Page 6: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

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

Page 7: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

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

Page 8: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

Your Turn Use logarithmic differentiation to

determine the derivative of

4

5

(3 1)( 2)

(2 5)( 6)

x xy

x x

Page 9: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

Why would you want to do that?

Reason #2Whenever you have an x in the exponent

3xy xy x

Page 10: Warm Up – NO CALCULATOR 1)Find the derivative of y = x 2 ln(x 3 ) 2)A particle moves along the x-axis so that at any time t > 0 its velocity is given by

You again!Find the derivative of

23xy x

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