BIJ Compute the derivative f'cts for 5./ fcto= log (x
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Title: Calculating Derivatives for Trigonometric and Logarithmic Functions**
**Objective:**
Learn how to compute the derivative of a function that involves both logarithmic and trigonometric components.
**Problem Statement:**
Compute the derivative, \( f'(x) \), for the function:
\[ f(x) = \log(x^4) \sin(x^3) \]
### Steps to Solve:
1. **Identify the Function Components:**
- Logarithmic part: \( \log(x^4) \)
- Trigonometric part: \( \sin(x^3) \)
2. **Apply the Product Rule:**
The product rule states that if you have a function \( u(x) \cdot v(x) \), its derivative is given by:
\[
(u \cdot v)' = u' \cdot v + u \cdot v'
\]
Here, let \( u(x) = \log(x^4) \) and \( v(x) = \sin(x^3) \).
3. **Differentiate Each Component:**
- **Differentiate the Logarithmic Part:**
\[
u(x) = \log(x^4) \Rightarrow u'(x) = \frac{d}{dx} \log(x^4)
\]
Using the chain rule, since \( \log(x^4) = 4\log(x) \):
\[
u'(x) = 4 \cdot \frac{1}{x} = \frac{4}{x}
\]
- **Differentiate the Trigonometric Part:**
\[
v(x) = \sin(x^3) \Rightarrow v'(x) = \cos(x^3) \cdot \frac{d}{dx}(x^3)
\]
Using the chain rule:
\[
v'(x) = \cos(x^3) \cdot 3x^2 = 3x^2 \cos(x^3)
\]
4. **Apply the Product Rule:**
Substitute the derivatives into the product rule:
\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
\[
f'(x) = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F75a69403-8651-4dbf-ac0e-b57dbd543e3d%2Fts8iq2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Derivatives for Trigonometric and Logarithmic Functions**
**Objective:**
Learn how to compute the derivative of a function that involves both logarithmic and trigonometric components.
**Problem Statement:**
Compute the derivative, \( f'(x) \), for the function:
\[ f(x) = \log(x^4) \sin(x^3) \]
### Steps to Solve:
1. **Identify the Function Components:**
- Logarithmic part: \( \log(x^4) \)
- Trigonometric part: \( \sin(x^3) \)
2. **Apply the Product Rule:**
The product rule states that if you have a function \( u(x) \cdot v(x) \), its derivative is given by:
\[
(u \cdot v)' = u' \cdot v + u \cdot v'
\]
Here, let \( u(x) = \log(x^4) \) and \( v(x) = \sin(x^3) \).
3. **Differentiate Each Component:**
- **Differentiate the Logarithmic Part:**
\[
u(x) = \log(x^4) \Rightarrow u'(x) = \frac{d}{dx} \log(x^4)
\]
Using the chain rule, since \( \log(x^4) = 4\log(x) \):
\[
u'(x) = 4 \cdot \frac{1}{x} = \frac{4}{x}
\]
- **Differentiate the Trigonometric Part:**
\[
v(x) = \sin(x^3) \Rightarrow v'(x) = \cos(x^3) \cdot \frac{d}{dx}(x^3)
\]
Using the chain rule:
\[
v'(x) = \cos(x^3) \cdot 3x^2 = 3x^2 \cos(x^3)
\]
4. **Apply the Product Rule:**
Substitute the derivatives into the product rule:
\[
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
\]
\[
f'(x) = \
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