# How do you differentiate f(x)=arctanx*3x using the product rule?

Nov 20, 2016

$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{3 x}{1 + {x}^{2}} + 3 \arctan x$

#### Explanation:

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Differentiating $f \left(x\right)$ is determined by applying product differentiation.
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Product Differentiation:
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$\textcolor{b l u e}{\frac{d}{\mathrm{dx}} \left(u \left(x\right) \times v \left(x\right)\right) = \frac{d}{\mathrm{dx}} u \left(x\right) \times v \left(x\right) + u \left(x\right) \times \frac{d}{\mathrm{dx}} v \left(x\right)}$
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$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left(\left(\arctan x\right) \times 3 x\right)$
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$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{d}{\mathrm{dx}} \left(\arctan x\right) \times 3 x + \arctan x \times \frac{d}{\mathrm{dx}} \left(3 x\right)$
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$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{1}{1 + {x}^{2}} \times 3 x + \arctan x \times 3$
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$\frac{d}{\mathrm{dx}} f \left(x\right) = \frac{3 x}{1 + {x}^{2}} + 3 \arctan x$