Key to "Use the rules for derivatives to find the..."

zi2lalZ

Answered question

2021-10-14

Use the rules for derivatives to find the derivative of function defined as follows.

y = \sqrt{2 t^{7} - 5}

ottcomn

Skilled2021-10-15Added 97 answers

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^{'} (x) = \frac{d}{d (g (x))} (f (x)) \cdot \frac{d}{d x} (g (x))

Applying the chain rule of derivative, we get

y^{'} = \frac{d y}{d t}

= \frac{d}{d t} (\sqrt{2 t^{7} - 5})

= \frac{d}{d (2 t^{7} - 5)} (\sqrt{2 t^{7} - 5}) \cdot \frac{d}{d t} (2 t^{7} - 5)

Step 2
To solve further, we use the power rule of differentiation,

\frac{d}{d x} (x^{n}) = n x^{n - 1}

y^{'} = \frac{d}{d (2 t^{7} - 5)} (\sqrt{2 t^{7} - 5}) \cdot \frac{d}{d t} (2 t^{7} - 5)

= (\frac{1}{2 \sqrt{2 t^{7} - 5}}) \cdot (2 (7 t^{6}) - 0)

= \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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