Derivative of $$$p \sqrt{v}$$$ with respect to $$$v$$$
The calculator will find the derivative of $$$p \sqrt{v}$$$ with respect to $$$v$$$, with steps shown.
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dv} \left(p \sqrt{v}\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{dv} \left(c f{\left(v \right)}\right) = c \frac{d}{dv} \left(f{\left(v \right)}\right)$$$ with $$$c = p$$$ and $$$f{\left(v \right)} = \sqrt{v}$$$:
$${\color{red}\left(\frac{d}{dv} \left(p \sqrt{v}\right)\right)} = {\color{red}\left(p \frac{d}{dv} \left(\sqrt{v}\right)\right)}$$Apply the power rule $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$ with $$$n = \frac{1}{2}$$$:
$$p {\color{red}\left(\frac{d}{dv} \left(\sqrt{v}\right)\right)} = p {\color{red}\left(\frac{1}{2 \sqrt{v}}\right)}$$Thus, $$$\frac{d}{dv} \left(p \sqrt{v}\right) = \frac{p}{2 \sqrt{v}}$$$.
Answer
$$$\frac{d}{dv} \left(p \sqrt{v}\right) = \frac{p}{2 \sqrt{v}}$$$A