328
Chapter 3 | Derivatives
dy dx at x =1. Using Equation 3.33 , we see that dy dx = 3 (3 x +1)ln2 .
To find the slope, we must evaluate
By evaluating the derivative at x =1, we see that the tangent line has slope dy dx | x =1 = 3 4ln2 = 3 ln16 .
Find the slope for the line tangent to y =3 x at x =2.
3.53
Logarithmic Differentiation At this point, we can take derivatives of functions of the form y = ⎛ ⎝ g ( x )
⎞ ⎠ n for certain values of n , as well as functions
of the form y = b g ( x ) , where b >0 and b ≠1. Unfortunately, we still do not know the derivatives of functions such as y = x x or y = x π . These functions require a technique called logarithmic differentiation , which allows us to differentiate any function of the form h ( x ) = g ( x ) f ( x ) . It can also be used to convert a very complex differentiation problem into a simpler one, such as finding the derivative of y = x 2 x +1 e x sin 3 x . We outline this technique in the following problem-solving strategy. Problem-Solving Strategy: Using Logarithmic Differentiation 1. To differentiate y = h ( x ) using logarithmic differentiation, take the natural logarithm of both sides of the equation to obtain ln y = ln ⎛ ⎝ h ( x ) ⎞ ⎠ . 2. Use properties of logarithms to expand ln ⎛ ⎝ h ( x ) ⎞ ⎠ as much as possible. 3. Differentiate both sides of the equation. On the left we will have 1 y dy dx . 4. Multiply both sides of the equation by y to solve for dy dx . 5. Replace y by h ( x ).
Example 3.81 Using Logarithmic Differentiation
tan x
Find the derivative of y = ⎛
⎝ 2 x 4 +1 ⎞ ⎠
.
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online