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

he298c 2021-10-04 Answered
Find the derivative using the appropriate rule or combination of rules.
\(\displaystyle{y}={\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}\)

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Expert Answer

Ayesha Gomez
Answered 2021-10-05 Author has 10704 answers
Step 1
We have to find the derivative of function:
\(\displaystyle{y}={\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}\)
We know the formula of derivatives,
\(\displaystyle{\frac{{{\left.{d}{x}\right.}^{{{n}}}}}{{{\left.{d}{x}\right.}}}}={n}{x}^{{{n}-{1}}}\)
\(\displaystyle{\frac{{{d}{a}{x}^{{{n}}}}}{{{\left.{d}{x}\right.}}}}={a}{\frac{{{\left.{d}{x}\right.}^{{{n}}}}}{{{\left.{d}{x}\right.}}}}\) (where a is constant)
\(\displaystyle{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{x}\right.}}}}={1}\)
\(\displaystyle{\frac{{{d}{a}}}{{{\left.{d}{x}\right.}}}}={0}\)
\(\displaystyle{\frac{{{d}{\left({f{{\left({x}\right)}}}\right)}^{{{n}}}}}{{{\left.{d}{x}\right.}}}}={n}{\left({f{{\left({x}\right)}}}\right)}^{{{n}-{1}}}{\frac{{{d}{f{{\left({x}\right)}}}}}{{{\left.{d}{x}\right.}}}}\)
Step 2
Differentiating the given function with respect to 't', we get
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}={\frac{{{d}{\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}{{{\left.{d}{t}\right.}}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}-{1}}}{\frac{{{d}{\left({4}{t}+{9}\right)}}}{{{\left.{d}{t}\right.}}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left({4}{t}+{9}\right)}^{{{\frac{{{1}-{2}}}{{{2}}}}}}{\left({4}{\frac{{{\left.{d}{t}\right.}}}{{{\left.{d}{t}\right.}}}}+{\frac{{{d}{9}}}{{{\left.{d}{t}\right.}}}}\right)}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{\left({4}{t}+{9}\right)}^{{-{\frac{{{1}}}{{{2}}}}}}{\left({4}\times{1}+{0}\right)}\)
\(\displaystyle={\frac{{{4}}}{{{2}}}}\times{\frac{{{1}}}{{{\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}}\)
\(\displaystyle={\frac{{{2}}}{{{\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}}\)
Hence, derivative of the function is \(\displaystyle{\frac{{{2}}}{{{\left({4}{t}+{9}\right)}^{{{\frac{{{1}}}{{{2}}}}}}}}}\).
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