zi2lalZ

2021-10-14

Use the rules for derivatives to find the derivative of function defined as follows.
$y=\sqrt{2{t}^{7}-5}$

ottcomn

Step 1
The given function is,
$y=\sqrt{2{t}^{7}-5}$
To find the derivative of the given function, we use the chain rule of derivative,
${f}^{\prime }\left(x\right)=\frac{d}{d\left(g\left(x\right)\right)}\left(f\left(x\right)\right)\cdot \frac{d}{dx}\left(g\left(x\right)\right)$
Applying the chain rule of derivative, we get
${y}^{\prime }=\frac{dy}{dt}$
$=\frac{d}{dt}\left(\sqrt{2{t}^{7}-5}\right)$
$=\frac{d}{d\left(2{t}^{7}-5\right)}\left(\sqrt{2{t}^{7}-5}\right)\cdot \frac{d}{dt}\left(2{t}^{7}-5\right)$
Step 2
To solve further, we use the power rule of differentiation, $\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$
${y}^{\prime }=\frac{d}{d\left(2{t}^{7}-5\right)}\left(\sqrt{2{t}^{7}-5}\right)\cdot \frac{d}{dt}\left(2{t}^{7}-5\right)$
$=\left(\frac{1}{2\sqrt{2{t}^{7}-5}}\right)\cdot \left(2\left(7{t}^{6}\right)-0\right)$
$=\frac{7{t}^{6}}{\sqrt{2{t}^{7}-5}}$
Therefore, the derivative of the given function is $\frac{7{t}^{6}}{\sqrt{2{t}^{7}-5}}$

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