Differentiate the function (x^2 + 3x) * e^(-x/2)
Understand the Problem
The question is asking us to perform various steps to differentiate a function that involves e raised to a power and polynomial terms. We need to follow the rules of differentiation, particularly the product rule and chain rule, to solve the provided expressions.
Answer
$$ \frac{dy}{dx} = -\frac{1}{2} e^{-\frac{x}{2}} (x - 3)(x + 2) $$
Answer for screen readers
The derivative of the function is $$ \frac{dy}{dx} = -\frac{1}{2} e^{-\frac{x}{2}} (x - 3)(x + 2) $$
Steps to Solve
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Identify the function and necessary rules We start with the function: $$ y = (x^2 + 3x) \cdot e^{-\frac{x}{2}} $$ We will use the product rule, which states that if $y = u \cdot v$, then $$ \frac{dy}{dx} = u'v + uv' $$
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Differentiate the components Let $$ u = x^2 + 3x \quad \text{and} \quad v = e^{-\frac{x}{2}} $$
First, we differentiate $u$: $$ u' = \frac{d}{dx}(x^2 + 3x) = 2x + 3 $$
Now, we differentiate $v$ using the chain rule: $$ v' = \frac{d}{dx}\left(e^{-\frac{x}{2}}\right) = e^{-\frac{x}{2}} \cdot \left(-\frac{1}{2}\right) = -\frac{1}{2}e^{-\frac{x}{2}} $$
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Apply the product rule Now substitute $u$, $u'$, $v$, and $v'$ into the product rule: $$ \frac{dy}{dx} = (2x + 3)e^{-\frac{x}{2}} + (x^2 + 3x)\left(-\frac{1}{2}e^{-\frac{x}{2}}\right) $$
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Factor out common terms We can factor out $e^{-\frac{x}{2}}$: $$ \frac{dy}{dx} = e^{-\frac{x}{2}} \left((2x + 3) - \frac{1}{2}(x^2 + 3x)\right) $$
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Simplify the expression Expand and combine like terms: $$ = e^{-\frac{x}{2}} \left((2x + 3) - \frac{1}{2}x^2 - \frac{3}{2}x\right) $$ $$ = e^{-\frac{x}{2}} \left(-\frac{1}{2}x^2 + \frac{1}{2}x + 3\right) $$
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Prepare for final expression We can express it more neatly: $$ = e^{-\frac{x}{2}}\left(-\frac{1}{2}(x^2 - x - 6)\right) $$
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Factor the quadratic Notice that $x^2 - x - 6$ factors to $(x - 3)(x + 2)$: $$ = -\frac{1}{2}e^{-\frac{x}{2}}(x - 3)(x + 2) $$
The derivative of the function is $$ \frac{dy}{dx} = -\frac{1}{2} e^{-\frac{x}{2}} (x - 3)(x + 2) $$
More Information
This derivative shows how the function changes with respect to $x$. The exponential decay represented by $e^{-\frac{x}{2}}$ indicates that the function will approach zero as $x$ increases.
Tips
- Forgetting to apply the product and chain rule correctly, especially when handling derivatives of exponential functions.
- Not simplifying the derivative expression fully before factoring.
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