Math Problem Statement
If y equals open parentheses cos space x close parentheses to the power of x squared end exponent comma spacethen fraction numerator d y over denominator d x end fraction equals
a. open parentheses cos space x close parentheses to the power of x squared end exponent open parentheses negative 2 x space tan space x space plus x squared space ln left parenthesis cos x right parenthesis close parentheses
b. x squared space open parentheses cos space x close parentheses to the power of x squared minus 1 end exponent open parentheses negative sin space x close parentheses left parenthesis 2 x right parenthesis
c. negative x squared space open parentheses cos space x close parentheses to the power of x squared end exponent open parentheses tan space x space minus ln left parenthesis cos space x right parenthesis close parentheses
d. negative x space open parentheses cos space x close parentheses to the power of x squared end exponent open parentheses x space tan space x space minus 2 space ln left parenthesis cos space x right parenthesis close parentheses
e. negative x space open parentheses cos space x close parentheses to the power of x squared end exponent open parentheses x space tan space x space plus 2 space ln left parenthesis cos space x right parenthesis close parentheses
Solution
To find where , we will use logarithmic differentiation.
Given:
-
Take the natural logarithm of both sides:
-
Differentiate both sides with respect to :
Using the chain rule on the left side and the product rule on the right side:
- The derivative of is:
Thus,
- Multiply both sides by to solve for :
Substitute :
Comparing this with the given options, we have:
a.
b.
c.
d.
e.
The correct answer matches option (a).
Would you like more details or have any questions? Here are 5 more questions you can ask:
- Can you explain the product rule in calculus?
- What is the chain rule and how is it used in differentiation?
- How do you find the derivative of a function raised to a power?
- What is logarithmic differentiation and when is it used?
- Can you solve another example involving logarithmic differentiation?
Tip: When solving complex differentiation problems, breaking them down using rules like the chain rule, product rule, and logarithmic differentiation can simplify the process and reduce errors.
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Math Problem Analysis
Mathematical Concepts
Logarithmic differentiation
Chain rule
Product rule
Formulas
\( \frac{d}{dx} (\ln u) = \frac{1}{u} \frac{du}{dx} \)
\( \frac{d}{dx} (uv) = u'v + uv' \)
Theorems
-
Suitable Grade Level
Advanced Mathematics