Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding the Derivative of a Composite Function
In this exercise, we are asked to find the derivative of the function:
\[ f(t) = \left(5t^3 - 4t^2 + 1\right)^{49} \]
To solve this, follow these steps:
1. **Identify the function inside the power function**:
- Here, the inner function is \( g(t) = 5t^3 - 4t^2 + 1 \).
- The outer function is \( h(u) = u^{49} \).
2. **Apply the chain rule**:
- The chain rule states that if we have a composite function \( f(x) = h(g(x)) \), the derivative is:
\[
f'(x) = h'(g(x)) \cdot g'(x)
\]
- For our function, this becomes:
\[
f'(t) = \frac{d}{dt} \left( (5t^3 - 4t^2 + 1)^{49} \right) = 49 \cdot (5t^3 - 4t^2 + 1)^{48} \cdot \frac{d}{dt} (5t^3 - 4t^2 + 1)
\]
3. **Find the derivative of the inner function \( g(t) \)**:
- The derivative of \( g(t) = 5t^3 - 4t^2 + 1 \) is:
\[
g'(t) = 15t^2 - 8t
\]
4. **Combine the results**:
- Substituting \( g'(t) \) into the chain rule result, we get:
\[
f'(t) = 49 \cdot (5t^3 - 4t^2 + 1)^{48} \cdot (15t^2 - 8t)
\]
Now you can input the final derivative \( f'(t) \) in the provided input box.
### Summary of Derivative:
\[
f'(t) = 49 \cdot (5t^3 - 4t^2 + 1)^{48} \cdot (15t^2 - 8t)
\]
This explicit representation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdaf30f56-016e-4284-9e0e-308d7a6d7bdd%2F5daea4ab-0ee3-4eaf-b58a-eb7a25f93db2%2Fcbs7y5c_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Derivative of a Composite Function
In this exercise, we are asked to find the derivative of the function:
\[ f(t) = \left(5t^3 - 4t^2 + 1\right)^{49} \]
To solve this, follow these steps:
1. **Identify the function inside the power function**:
- Here, the inner function is \( g(t) = 5t^3 - 4t^2 + 1 \).
- The outer function is \( h(u) = u^{49} \).
2. **Apply the chain rule**:
- The chain rule states that if we have a composite function \( f(x) = h(g(x)) \), the derivative is:
\[
f'(x) = h'(g(x)) \cdot g'(x)
\]
- For our function, this becomes:
\[
f'(t) = \frac{d}{dt} \left( (5t^3 - 4t^2 + 1)^{49} \right) = 49 \cdot (5t^3 - 4t^2 + 1)^{48} \cdot \frac{d}{dt} (5t^3 - 4t^2 + 1)
\]
3. **Find the derivative of the inner function \( g(t) \)**:
- The derivative of \( g(t) = 5t^3 - 4t^2 + 1 \) is:
\[
g'(t) = 15t^2 - 8t
\]
4. **Combine the results**:
- Substituting \( g'(t) \) into the chain rule result, we get:
\[
f'(t) = 49 \cdot (5t^3 - 4t^2 + 1)^{48} \cdot (15t^2 - 8t)
\]
Now you can input the final derivative \( f'(t) \) in the provided input box.
### Summary of Derivative:
\[
f'(t) = 49 \cdot (5t^3 - 4t^2 + 1)^{48} \cdot (15t^2 - 8t)
\]
This explicit representation
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