Answered: "Find the derivatives of the functions. r(t)=log3(t+t)"

Alyce Wilkinson

Answered question

2021-06-23

Find the derivatives of the functions.

r (t) = \log_{3} (t + \sqrt{t})

Laith Petty

Skilled2021-06-24Added 103 answers

r (t) = \log_{3} (t + \sqrt{t})

Rewrite the function, recall that

\sqrt{a} = a^{\frac{1}{2}}

, so

r (t) = \log_{3} (t + t^{\frac{1}{2}})

Diffetrentiate both sides with respect to x

r^{'} (t) = \frac{d}{d t} [\log_{3} (t + t^{\frac{1}{2}})]

Apply

\frac{d}{d t} [\log_{b}^{| u |}] = \frac{1}{u \ln b} {\frac{d u}{d t}

, let

u = t + t^{\frac{1}{2}}

, so

r^{'} (t) = \frac{1}{(t + t^{\frac{1}{2}}) \ln 3} \frac{d}{d t} [t + t^{\frac{1}{2}}]

Therefore,

r^{'} (t) = \frac{1}{(t + t^{\frac{1}{2}}) \ln 3} (1 + \frac{1}{2} t^{- \frac{1}{2}})

Simplify

r^{'} (t) = \frac{1 + \frac{1}{2} t^{- \frac{1}{2}}}{(t + t^{\frac{1}{2}}) \ln 3}

r^{'} (t) = \frac{1 + \frac{1}{2 \sqrt{t}}}{(t + \sqrt{t}) \ln 3}

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