3.8: Homework- Product Rule
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Take the derivatives of the following functions.
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\(f(x) = x^2 e^x\)
\(f'(x) = x^2 e^x + 2x e^x\)ans
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\(g(x) = \left(\sqrt{x} + 1\right)\left(x^2\right)\)
ans
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\(h(x) = \frac{1}{x} \left(e^x + 1 \right)\)
\(h'(x) = -\frac{1}{x^2} (e^x + 1) + \frac{1}{x} e^x\)ans
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\(i(x) = \ln(x) x\)
\(i'(x) = 1 + \ln(x)\)ans
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\(j(x) = \sqrt{x} \ln(x)\)
\(j(x) = \frac{1}{2 \sqrt{x}} \ln(x) + \frac{1}{\sqrt{x}}\)ans
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\(m(x) = (x^3 + x^2) \sin(x)\)
\(m'(x) = (3x^2 + 2x) \sin(x) + (x^3 + x^2) \cos(x)\)ans
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\(l(x) = e^{2x}\) (Hint: You can rewrite this as \(e^x \cdot e^x\))
\(l'(x) = 2 e^{2x}\)ans
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\(\ell(x) = x e^x \ln(x)\)
\(\ell'(x) = e^x \ln(x) + e^x \ln(x) + e^x\)ans
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\(f(x) = x^2 e^x\)