"Find the derivatives of the functionsg(x)=tan⁡3x(x+7)4" was answered by our experts

mwombenizhjb

Answered question

2022-03-25

Find the derivatives of the functions

g (x) = \frac{\tan 3 x}{{(x + 7)}^{4}}

alwadau8ndv

Beginner2022-03-26Added 9 answers

Step 1
Differentiation of the rational function can be done with the help of the differentiation formula

\frac{d (\frac{u}{v})}{d x} = \frac{v \frac{d u}{d x} - u \frac{d v}{d x}}{v^{2}}

.
Differentiate the function with respect to x to find the required derivative.
Step 2
Differentiate the function

g (x) = \frac{\tan 3 x}{{(x + 7)}^{4}}

with the help of the above differentiation formula.
Apply the differentiation formula

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

at the time of differentiation.

\frac{d (g (x))}{d x} = \frac{d (\frac{\tan 3 x}{{(x + 7)}^{4}})}{d x}

= \frac{{(x + 7)}^{4} \frac{d (\tan x)}{d x} - \tan 3 x \frac{d ({(x + 7)}^{4})}{d x}}{{({(x + 7)}^{4})}^{2}}

= \frac{3 x \sec^{2} (3 x) + 21 \sec^{2} (3 x) - 4 \tan (3 x)}{{(x + 7)}^{5}}

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