Nate tried to find the derivative of $\left(x^{2}+5 x\right) \cdot \sin (x)$. Here is his work:
Step 1: This is the product of $x^{2}+5 x$ and $\sin (x)$. So we should use the product rule.

Step 2:
$$
\frac{d}{d x}\left[\left(x^{2}+5 x\right) \cdot \sin (x)\right]=\frac{d}{d x}\left[x^{2}+5 x\right] \cdot \frac{d}{d x}[\sin (x)]
$$

Step 3: Finding the derivatives of the factors:
$$
\begin{array}{c}
\frac{d}{d x}\left[x^{2}+5 x\right]=2 x+5 \
\frac{d}{d x}[\sin (x)]=\cos (x)
\end{array}
$$

Step 4: Putting it all together:
$$
\begin{aligned}
& \frac{d}{d x}\left[\left(x^{2}+5 x\right) \cdot \sin (x)\right] \
= & \frac{d}{d x}\left[x^{2}+5 x\right] \cdot \frac{d}{d x}[\sin (x)] \
= & (2 x+5) \cdot \cos (x)
\end{aligned}
$$

Is Nate's work correct? If not, what's his mistake?
Choose 1 answer:
(A) Nate's work is correct.
(B) Step 1 is incorrect. Nate should have used a different rule and not the product rule.
(C) Step 2 is incorrect. Nate didn't state the correct product rule.
(D) Step 3 is incorrect. Nate didn't differentiate $\sin (x)$ correctly.

Question

Nate tried to find the derivative of $\left(x^{2}+5 x\right) \cdot \sin (x)$. Here is his work:
Step 1: This is the product of $x^{2}+5 x$ and $\sin (x)$. So we should use the product rule.

Step 2:
$$
\frac{d}{d x}\left[\left(x^{2}+5 x\right) \cdot \sin (x)\right]=\frac{d}{d x}\left[x^{2}+5 x\right] \cdot \frac{d}{d x}[\sin (x)]
$$

Step 3: Finding the derivatives of the factors:
$$
\begin{array}{c}
\frac{d}{d x}\left[x^{2}+5 x\right]=2 x+5 \
\frac{d}{d x}[\sin (x)]=\cos (x)
\end{array}
$$

Step 4: Putting it all together:
$$
\begin{aligned}
& \frac{d}{d x}\left[\left(x^{2}+5 x\right) \cdot \sin (x)\right] \
= & \frac{d}{d x}\left[x^{2}+5 x\right] \cdot \frac{d}{d x}[\sin (x)] \
= & (2 x+5) \cdot \cos (x)
\end{aligned}
$$

Is Nate's work correct? If not, what's his mistake?
Choose 1 answer:
(A) Nate's work is correct.
(B) Step 1 is incorrect. Nate should have used a different rule and not the product rule.
(C) Step 2 is incorrect. Nate didn't state the correct product rule.
(D) Step 3 is incorrect. Nate didn't differentiate $\sin (x)$ correctly.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Nate correctly identifies that the expression $(x^2 + 5x) \cdot \sin(x)$ is a product and that the product rule should be used. ‖ ‖ step 2 ⋮ Nate incorrectly states the product rule in Step 2. The correct product rule is $\frac{d}{dx}[u \cdot v] = u' \cdot v + u \cdot v'$, where $u = x^2 + 5x$ and $v = \sin(x)$. ‖ ‖ step 3 ⋮ Nate correctly finds the derivatives of the factors: $\frac{d}{dx}[x^2 + 5x] = 2x + 5$ and $\frac{d}{dx}[\sin(x)] = \cos(x)$. ‖ ‖ step 4 ⋮ Nate incorrectly combines the derivatives in Step 4. The correct combination using the product rule is: $$ \frac{d}{dx}[(x^2 + 5x) \cdot \sin(x)] = (2x + 5) \cdot \sin(x) + (x^2 + 5x) \cdot \cos(x) $$ ‖ ***C*** ∻Key Concept∻ ⚹Product Rule⚹ ∻Explanation∻ ⚹The product rule for differentiation states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.⚹◊What is the correct product rule in calculus that Nate should have used in Step 2 of his work⍭ Generate me a similar question◊

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