Logarithmic Differentiation Calculator

Calculate derivatives step by step using logarithms

The online calculator will calculate the derivative of any function using the logarithmic differentiation, with steps shown. Also, it will evaluate the derivative at the given point if needed.

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

Solution

Let $$$H{\left(x \right)} = x^{\sin{\left(x \right)}}$$$.

Take the logarithm of both sides: $$$\ln{\left(H{\left(x \right)} \right)} = \ln{\left(x^{\sin{\left(x \right)}} \right)}$$$.

Rewrite the RHS using the properties of logarithms: $$$\ln{\left(H{\left(x \right)} \right)} = \ln{\left(x \right)} \sin{\left(x \right)}$$$.

Differentiate separately both sides of the equation: $$$\frac{d}{dx} \left(\ln{\left(H{\left(x \right)} \right)}\right) = \frac{d}{dx} \left(\ln{\left(x \right)} \sin{\left(x \right)}\right)$$$.

Differentiate the LHS of the equation.

The function $$$\ln{\left(H{\left(x \right)} \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \ln{\left(u \right)}$$$ and $$$g{\left(x \right)} = H{\left(x \right)}$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln{\left(H{\left(x \right)} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln{\left(u \right)}\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$

The derivative of the natural logarithm is $$$\frac{d}{du} \left(\ln{\left(u \right)}\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln{\left(u \right)}\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$

Return to the old variable:

$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$

Thus, $$$\frac{d}{dx} \left(\ln{\left(H{\left(x \right)} \right)}\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.

Differentiate the RHS of the equation.

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 $$$f{\left(x \right)} = \ln{\left(x \right)}$$$ and $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:

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

The derivative of the natural logarithm is $$$\frac{d}{dx} \left(\ln{\left(x \right)}\right) = \frac{1}{x}$$$:

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

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

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

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

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

Therefore, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(\ln{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) H{\left(x \right)} = x^{\sin{\left(x \right)} - 1} \left(x \ln{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right).$$$

Answer

$$$\frac{d}{dx} \left(x^{\sin{\left(x \right)}}\right) = x^{\sin{\left(x \right)} - 1} \left(x \ln{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)}\right)$$$A