Question
Use product Rule to find \( f^{\prime}(x) \) where \( f(x)=(x+2)\left(3 x^{3}-5 x\right) \) Type your answer starting with the term with highest degree \( \square \)
Ask by Perkins Schneider. in the United States
Feb 10,2025
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Tutor-Verified Answer
Answer
\( f^{\prime}(x) = 12x^{3} + 18x^{2} - 10x - 10 \)
Solution
To differentiate f(x) = (x + 2)(3x³ - 5x) using the product rule, let
u(x) = x + 2 and v(x) = 3x³ - 5x.
Then, u'(x) = 1 and v'(x) = 9x² - 5.
According to the product rule, f'(x) = u'(x)·v(x) + u(x)·v'(x). Substituting in:
f'(x) = 1·(3x³ - 5x) + (x + 2)·(9x² - 5)
= 3x³ - 5x + (9x³ + 18x² - 5x - 10)
= (3x³ + 9x³) + 18x² + (-5x - 5x) - 10
= 12x³ + 18x² - 10x - 10.
Thus, the derivative with the term of highest degree first is:
12x³ + 18x² - 10x - 10.
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Simplify this solution The Deep Dive
To find the derivative \( f^{\prime}(x) \) using the product rule, we first set \( u = x + 2 \) and \( v = 3x^3 - 5x \). The product rule states: \[ f^{\prime}(x) = u'v + uv' \] Calculating \( u' \) and \( v' \): \[ u' = 1, \quad v' = 9x^2 - 5 \] Now, applying the product rule: \[ f^{\prime}(x) = (1)(3x^3 - 5x) + (x + 2)(9x^2 - 5) \] Simplifying: \[ f^{\prime}(x) = 3x^3 - 5x + (x + 2)(9x^2 - 5) \] Now, multiply out \( (x + 2)(9x^2 - 5) \): \[ = 9x^3 - 5x + 18x^2 - 10 \] Combine everything: \[ = 3x^3 - 5x + 9x^3 + 18x^2 - 10 - 5x \] \[ = (3x^3 + 9x^3) + 18x^2 + (-5x - 5x) - 10 \] \[ = 12x^3 + 18x^2 - 10 \] So the final answer is: \[ f^{\prime}(x) = 12x^3 + 18x^2 - 10 \]
