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he298c

Answered question

2021-10-04

Find the derivative using the appropriate rule or combination of rules.

y = {(4 t + 9)}^{\frac{1}{2}}

Ayesha Gomez

Skilled2021-10-05Added 104 answers

Step 1
We have to find the derivative of function:

y = {(4 t + 9)}^{\frac{1}{2}}

We know the formula of derivatives,

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

\frac{d a x^{n}}{d x} = a \frac{{d x}^{n}}{d x}

(where a is constant)

\frac{d x}{d x} = 1

\frac{d a}{d x} = 0

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

Step 2
Differentiating the given function with respect to t, we get

\frac{d y}{d t} = \frac{d {(4 t + 9)}^{\frac{1}{2}}}{d t}

= \frac{1}{2} {(4 t + 9)}^{\frac{1}{2} - 1} \frac{d (4 t + 9)}{d t}

= \frac{1}{2} {(4 t + 9)}^{\frac{1 - 2}{2}} (4 \frac{d t}{d t} + \frac{d 9}{d t})

= \frac{1}{2} {(4 t + 9)}^{- \frac{1}{2}} (4 \times 1 + 0)

= \frac{4}{2} \times \frac{1}{{(4 t + 9)}^{\frac{1}{2}}}

= \frac{2}{{(4 t + 9)}^{\frac{1}{2}}}

Hence, derivative of the function is

\frac{2}{{(4 t + 9)}^{\frac{1}{2}}}

Jeffrey Jordon

Expert2022-02-01Added 2605 answers

Answer is given below (on video)

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