# Online Derivative Calculator with Steps

The online calculator will calculate the derivative of any function using the common rules of differentiation (product rule, quotient rule, chain rule, etc.), with steps shown. It can handle polynomial, rational, irrational, exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions. Also, it will evaluate the derivative at the given point if needed. It also supports computing the first, second, and third derivatives, up to 10.

Related calculator: Implicit Differentiation Calculator with Steps

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Find $\frac{d}{dx} \left(x \sin{\left(x \right)}\right)$.

## Solution

Apply the product rule $\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$ with $f{\left(x \right)} = x$ and $g{\left(x \right)} = \sin{\left(x \right)}$:

$$\color{red}{\left(\frac{d}{dx} \left(x \sin{\left(x \right)}\right)\right)} = \color{red}{\left(\frac{d}{dx} \left(x\right) \sin{\left(x \right)} + x \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$

Apply the power rule $\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$ with $n = 1$, in other words, $\frac{d}{dx} \left(x\right) = 1$:

$$x \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \sin{\left(x \right)} \color{red}{\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \sin{\left(x \right)} \color{red}{\left(1\right)}$$

The derivative of the sine is $\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$:

$$x \color{red}{\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \sin{\left(x \right)} = x \color{red}{\left(\cos{\left(x \right)}\right)} + \sin{\left(x \right)}$$

Thus, $\frac{d}{dx} \left(x \sin{\left(x \right)}\right) = x \cos{\left(x \right)} + \sin{\left(x \right)}$.

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