What is the tangent of f(x)=ln((7(x+3))/x) ?

The tangent of f(x)=ln((7(x+3))/x) is y=-3/(a_{0)(a_{0}+3)}x+ln((7(a_{0}+3))/(a_{0)})+3/(a_{0)+3}

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tangent of f(x)=ln((7(x+3))/x)

tangent of

f (x) = ln (\frac{7 ( x + 3 )}{x})

y = - \frac{3}{a _{0} ( a _{0} + 3 )} x + ln (\frac{7 ( a _{0} + 3 )}{a _{0}}) + \frac{3}{a _{0} + 3}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, ln (\frac{7 ( a _{0} + 3 )}{a _{0}}))

Find the slope of

f (x) = ln (\frac{7 ( x + 3 )}{x}) : \frac{df ( x )}{d x} = - \frac{3}{x ( x + 3 )}

EN : T i tl e G e n er a l Eq u a t i o n Sl o p e A t P o in t 2 Eq : m = - \frac{3}{a _{0} ( a _{0} + 3 )}

Find the line with slope m=

- \frac{3}{a _{0} ( a _{0} + 3 )}

and passing through

(a_{0}, ln (\frac{7 ( a _{0} + 3 )}{a _{0}})) : y = - \frac{3}{a _{0} ( a _{0} + 3 )} x + ln (\frac{7 ( a _{0} + 3 )}{a _{0}}) + \frac{3}{a _{0} + 3}

y = - \frac{3}{a _{0} ( a _{0} + 3 )} x + ln (\frac{7 ( a _{0} + 3 )}{a _{0}}) + \frac{3}{a _{0} + 3}

What is the tangent of f(x)=ln((7(x+3))/x) ?
The tangent of f(x)=ln((7(x+3))/x) is y=-3/(a_{0)(a_{0}+3)}x+ln((7(a_{0}+3))/(a_{0)})+3/(a_{0)+3}