Chapter 3 | Derivatives
329
Solution Use logarithmic differentiation to find this derivative. ln y = ln ⎛ ⎝ 2 x 4 +1 ⎞ ⎠ tan x
Step 1. Take the natural logarithm of both sides. Step 2. Expand using properties of logarithms.
ln y = tan x ln ⎛
⎝ 2 x 4 +1 ⎞ ⎠
Step 3. Differentiate both sides. Use the product rule on the right.
3 2 x 4 +1 ⎠ + 8 x
dy dx = sec
⎛ ⎝ 2 x 4 +1 ⎞
⎠ + 8 x
1 y
2 x ln
· tan x
⎛ ⎝ sec 2 x ln
⎞ ⎠
3 2 x 4 +1 ⎛ ⎝ 2 x 4 +1 ⎞
dy dx = dy dx =
⎛ ⎝ 2 x 4 +1 ⎞
y ·
· tan x
Step 4. Multiply by y on both sides.
tan x ⎛
⎞ ⎠ Step 5. Substitute y = ⎛
tan x
3 2 x 4 +1
⎛ ⎝ 2 x 4 +1 ⎞ ⎠
⎝ 2 x 4 +1 ⎞ ⎠
⎠ + 8 x
⎝ sec 2 x ln
· tan x
.
Example 3.82 Using Logarithmic Differentiation
Find the derivative of y = x 2 x +1 e x sin 3 x .
Solution This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter. ln y = ln x 2 x +1 e x sin 3 x Step 1. Take the natural logarithm of both sides. ln y = ln x + 1 2 ln (2 x +1)− x ln e −3lnsin x Step 2. Expand using properties of logarithms. 1 y dy dx = 1 x + 1 2 x +1 −1−3 cos x sin x Step 3. Differentiate both sides.
y ⎛
x ⎞ ⎠
dy dx = dy dx =
⎝ 1 x + 1 2 x +1 −1−3cot
Step 4. Multiply by y on both sides. Step 5. Substitute y = x 2 x +1 e x sin 3 x .
⎛ ⎝ 1 x + 1 2 x +1 −1−3cot
x ⎞ ⎠
x 2 x +1 e x sin 3 x
Example 3.83 Extending the Power Rule Find the derivative of y = x r where r is an arbitrary real number.
Solution The process is the same as in Example 3.82 , though with fewer complications.
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