# How do you differentiate f(x)=sqrtxsinx?

Jun 14, 2018

$\frac{d}{\mathrm{dx}} \left(\sqrt{x} \sin \left(x\right)\right) = \sin \frac{x}{2 \sqrt{x}} + \sqrt{x} \cos \left(x\right)$

#### Explanation:

To take the derivative of this, we need two rules. The first rule is the chain rule, which states:

$\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$

The second rule we need is the power rule, which states:

$\frac{d}{\mathrm{dx}} \left({x}^{n}\right) = n {x}^{n - 1}$

Now, we can start taking the derivative. To simplify this, I'm going to rewrite $\sqrt{x}$ as ${x}^{\frac{1}{2}}$, which is perfectly valid.

To start, I'm going to take the derivative of each section.

$\frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2}}\right) = \frac{1}{2} {x}^{- \frac{1}{2}}$

$\frac{d}{\mathrm{dx}} \left({x}^{\frac{1}{2}}\right) = \frac{1}{2 \sqrt{x}}$

$\frac{d}{\mathrm{dx}} \left(\sin \left(x\right)\right) = \cos \left(x\right)$

Now that we have them, we can put them together:

$\frac{d}{\mathrm{dx}} \left(\sqrt{x} \sin \left(x\right)\right) = \sin \frac{x}{2 \sqrt{x}} + \sqrt{x} \cos \left(x\right)$