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Derivative of $$$\frac{\sqrt{2}}{2 \sqrt{t}}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dt} \left(\frac{\sqrt{2}}{2 \sqrt{t}}\right)$$$.
Solution
Apply the constant multiple rule $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ with $$$c = \frac{\sqrt{2}}{2}$$$ and $$$f{\left(t \right)} = \frac{1}{\sqrt{t}}$$$:
$${\color{red}\left(\frac{d}{dt} \left(\frac{\sqrt{2}}{2 \sqrt{t}}\right)\right)} = {\color{red}\left(\frac{\sqrt{2}}{2} \frac{d}{dt} \left(\frac{1}{\sqrt{t}}\right)\right)}$$Apply the power rule $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ with $$$n = - \frac{1}{2}$$$:
$$\frac{\sqrt{2} {\color{red}\left(\frac{d}{dt} \left(\frac{1}{\sqrt{t}}\right)\right)}}{2} = \frac{\sqrt{2} {\color{red}\left(- \frac{1}{2 t^{\frac{3}{2}}}\right)}}{2}$$Thus, $$$\frac{d}{dt} \left(\frac{\sqrt{2}}{2 \sqrt{t}}\right) = - \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}$$$.
Answer
$$$\frac{d}{dt} \left(\frac{\sqrt{2}}{2 \sqrt{t}}\right) = - \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}$$$A