Derivative of $$$- \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dt} \left(- \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}\right)$$$.
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 $$$c = - \frac{\sqrt{2}}{4}$$$ and $$$f{\left(t \right)} = \frac{1}{t^{\frac{3}{2}}}$$$:
$${\color{red}\left(\frac{d}{dt} \left(- \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}\right)\right)} = {\color{red}\left(- \frac{\sqrt{2}}{4} \frac{d}{dt} \left(\frac{1}{t^{\frac{3}{2}}}\right)\right)}$$Apply the power rule $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ with $$$n = - \frac{3}{2}$$$:
$$- \frac{\sqrt{2} {\color{red}\left(\frac{d}{dt} \left(\frac{1}{t^{\frac{3}{2}}}\right)\right)}}{4} = - \frac{\sqrt{2} {\color{red}\left(- \frac{3}{2 t^{\frac{5}{2}}}\right)}}{4}$$Thus, $$$\frac{d}{dt} \left(- \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}\right) = \frac{3 \sqrt{2}}{8 t^{\frac{5}{2}}}$$$.
Answer
$$$\frac{d}{dt} \left(- \frac{\sqrt{2}}{4 t^{\frac{3}{2}}}\right) = \frac{3 \sqrt{2}}{8 t^{\frac{5}{2}}}$$$A