Chapter2: Functions And Their Graphs
Section2.4: A Library Of Parent Functions
Problem 47E: During a nine-hour snowstorm, it snows at a rate of 1 inch per hour for the first 2 hours, at a rate...
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![### Problem Statement
Let \( y = f(x) \), where
\[ f(x) = 7x^3 + 3x. \]
Find the differential of the function.
\[ dy = \boxed{} \]
(Note: There is an incorrect symbol or indication in red that appears to signify a wrong answer or an error.)
### Explanation
To find the differential of the function \( f(x) \), we will first need to compute the derivative of \( f(x) \).
\[ f(x) = 7x^3 + 3x \]
Taking the derivative \( f'(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(7x^3) + \frac{d}{dx}(3x) \]
\[ f'(x) = 7 \cdot 3x^{3-1} + 3 \cdot 1 \]
\[ f'(x) = 21x^2 + 3 \]
The differential \( dy \) can now be found using:
\[ dy = f'(x) \, dx \]
\[ dy = (21x^2 + 3) \, dx \]
This result should be placed inside the provided box for the final answer:
\[ dy = \boxed{21x^2 + 3 \, dx} \]
(Note: The red 'X' might indicate an incorrect previous attempt or response.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54a9d33b-c075-4af5-8013-3f49004b10ba%2F35609383-f3e6-4826-bead-c14b443aabf6%2Fpe6m24s_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Let \( y = f(x) \), where
\[ f(x) = 7x^3 + 3x. \]
Find the differential of the function.
\[ dy = \boxed{} \]
(Note: There is an incorrect symbol or indication in red that appears to signify a wrong answer or an error.)
### Explanation
To find the differential of the function \( f(x) \), we will first need to compute the derivative of \( f(x) \).
\[ f(x) = 7x^3 + 3x \]
Taking the derivative \( f'(x) \) with respect to \( x \):
\[ f'(x) = \frac{d}{dx}(7x^3) + \frac{d}{dx}(3x) \]
\[ f'(x) = 7 \cdot 3x^{3-1} + 3 \cdot 1 \]
\[ f'(x) = 21x^2 + 3 \]
The differential \( dy \) can now be found using:
\[ dy = f'(x) \, dx \]
\[ dy = (21x^2 + 3) \, dx \]
This result should be placed inside the provided box for the final answer:
\[ dy = \boxed{21x^2 + 3 \, dx} \]
(Note: The red 'X' might indicate an incorrect previous attempt or response.)
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