Question
On the left-hand slde, for the first term \( 9 x^{3} \), we have the following. \( \frac{d}{d x}\left[9 x^{3}\right]=\square \) Now, note that the second term \( x^{2} y \) is a product, so we must use the product rule. We therefore have the following \( \frac{d}{d x}\left[x^{2} y\right]=\left(x^{2}\right)\left(y^{\prime}\right)+\square \)
Ask by Dawson Huff. in the United States
Mar 20,2025
Upstudy AI Solution
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Answer
The derivative of \(9x^{3}\) is \(27x^{2}\). For \(x^{2}y\), using the product rule, the derivative is \(x^{2}y' + 2xy\).
Solution
The derivative of the first term is
\[
\frac{d}{d x}\left[9 x^{3}\right]=27x^{2}.
\]
For the second term, using the product rule we differentiate \(x^{2}\) (yielding \(2x\)) and multiply by \(y\), then add \(x^{2}\) multiplied by the derivative of \(y\) (written as \(y'\)). Thus, we have
\[
\frac{d}{d x}\left[x^{2} y\right]=x^{2}y^{\prime}+2xy.
\]
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The derivative of the term \( 9x^3 \) is straightforward! When you apply the power rule, you multiply the exponent by the coefficient and decrease the exponent by one. So, \( \frac{d}{d x}\left[9 x^{3}\right]=27 x^{2} \). For the product rule, you want to differentiate each part of the product individually. The complete expression becomes \( \frac{d}{d x}\left[x^{2} y\right]=\left(x^{2}\right)\left(y^{\prime}\right)+\left(2x\right)\left(y\right) \). This means you differentiate \( x^2 \) to get \( 2x \) and multiply it by \( y \), adding that to the original \( x^2 \) multiplied by the derivative of \( y \).
