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Answer
The derivative of \( y = x^9 e^x \) is \( y' = x^8 e^x (9 + x) \).
Solution
Let \( u = x^9 \) and \( v = e^x \). Then we have:
\[
y = u \cdot v = x^9 e^x
\]
Using the product rule, which states:
\[
\frac{d}{dx}(uv) = u'v + uv'
\]
we first compute the derivatives of \( u \) and \( v \):
\[
u' = \frac{d}{dx}(x^9) = 9x^8
\]
\[
v' = \frac{d}{dx}(e^x) = e^x
\]
Substitute these into the product rule:
\[
y' = 9x^8 e^x + x^9 e^x
\]
Factor out the common factors \( x^8 e^x \):
\[
y' = x^8 e^x (9 + x)
\]
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Simplify this solution Mind Expander
To differentiate the function \( y = x^{9} e^{x} \), we need to apply the product rule, which states that if you have two functions \( u \) and \( v \), then the derivative \( (uv)' = u'v + uv' \). Here, let \( u = x^{9} \) and \( v = e^{x} \). First, we find the derivatives: - \( u' = 9x^{8} \) - \( v' = e^{x} \) Now, applying the product rule: \[ \frac{dy}{dx} = u'v + uv' = (9x^{8})(e^{x}) + (x^{9})(e^{x}) \] This simplifies to: \[ \frac{dy}{dx} = e^{x}(9x^{8} + x^{9}) \] Therefore, the final answer is: \[ \frac{dy}{dx} = e^{x}(9x^{8} + x^{9}) \]
