2 Find the derivative of the function. f(x) = (2x³ - 5x² + 4)³ f(x) = √√√5x + 1 g(t) ƒ(0) = = 1 (2t + 1)² cos(02) 8. f(x) = (x³ + 3x² - x) 50 10. f(x) = 12. F(t) = 1 x² - 1 1 2t + 1 14. g(0) = cos² A
2 Find the derivative of the function. f(x) = (2x³ - 5x² + 4)³ f(x) = √√√5x + 1 g(t) ƒ(0) = = 1 (2t + 1)² cos(02) 8. f(x) = (x³ + 3x² - x) 50 10. f(x) = 12. F(t) = 1 x² - 1 1 2t + 1 14. g(0) = cos² A
Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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Need help with #12 please.
![### Find the Derivative of the Function
#### Examples:
1. **Function:**
\[
f(x) = (2x^3 - 5x^2 + 4)^5
\]
2. **Function:**
\[
f(x) = \sqrt{5x + 1}
\]
3. **Function:**
\[
g(t) = \frac{1}{(2t + 1)^2}
\]
4. **Function:**
\[
f(\theta) = \cos(\theta^2)
\]
5. **Function:**
\[
f(x) = (x^5 + 3x^2 - x)^{50}
\]
6. **Function:**
\[
f(x) = \frac{1}{\sqrt[3]{x^2 - 1}}
\]
7. **Function:**
\[
F(t) = \left( \frac{1}{2t + 1} \right)^4
\]
8. **Function:**
\[
g(\theta) = \cos^2(\theta)
\]
These functions represent a variety of types, including polynomial expressions raised to a power, square roots, rational functions, trigonometric functions, and combinations thereof. The goal is to find the derivative for each function using the appropriate calculus rules, such as the power rule, chain rule, quotient rule, and trigonometric derivatives.
For example, for the first function \( f(x) = (2x^3 - 5x^2 + 4)^5 \):
- Apply the **chain rule**: If \( u(x) = 2x^3 - 5x^2 + 4 \), then \( f(x) = u(x)^5 \). The derivative \( f'(x) = 5 \cdot u(x)^4 \cdot u'(x) \).
For the second function \( f(x) = \sqrt{5x + 1} \):
- Express the square root as a power: \( f(x) = (5x + 1)^{1/2} \), then use the **chain rule** to differentiate.
Each function requires careful application of these differentiation techniques to find the correct derivative.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe88ba205-1ef5-4c9b-bc8a-d6f3eacaf2aa%2Fd9aa1f90-d88c-4c21-9f0b-fa971eff57b3%2F76uqtlj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Find the Derivative of the Function
#### Examples:
1. **Function:**
\[
f(x) = (2x^3 - 5x^2 + 4)^5
\]
2. **Function:**
\[
f(x) = \sqrt{5x + 1}
\]
3. **Function:**
\[
g(t) = \frac{1}{(2t + 1)^2}
\]
4. **Function:**
\[
f(\theta) = \cos(\theta^2)
\]
5. **Function:**
\[
f(x) = (x^5 + 3x^2 - x)^{50}
\]
6. **Function:**
\[
f(x) = \frac{1}{\sqrt[3]{x^2 - 1}}
\]
7. **Function:**
\[
F(t) = \left( \frac{1}{2t + 1} \right)^4
\]
8. **Function:**
\[
g(\theta) = \cos^2(\theta)
\]
These functions represent a variety of types, including polynomial expressions raised to a power, square roots, rational functions, trigonometric functions, and combinations thereof. The goal is to find the derivative for each function using the appropriate calculus rules, such as the power rule, chain rule, quotient rule, and trigonometric derivatives.
For example, for the first function \( f(x) = (2x^3 - 5x^2 + 4)^5 \):
- Apply the **chain rule**: If \( u(x) = 2x^3 - 5x^2 + 4 \), then \( f(x) = u(x)^5 \). The derivative \( f'(x) = 5 \cdot u(x)^4 \cdot u'(x) \).
For the second function \( f(x) = \sqrt{5x + 1} \):
- Express the square root as a power: \( f(x) = (5x + 1)^{1/2} \), then use the **chain rule** to differentiate.
Each function requires careful application of these differentiation techniques to find the correct derivative.
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