What is the tangent of 1/(2sqrt(x+1)),\at a=3 ?

The tangent of 1/(2sqrt(x+1)),\at a=3 is y=-1/(4(a+1)^{3/2)}x+(3a+2)/(4(a+1)^{3/2)}

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tangent of 1/(2sqrt(x+1)),\at a=3

tangent of

\frac{1}{2 x + 1}, at a = 3

y = - \frac{1}{4 ( a + 1 ) ^{\frac{3}{2}}} x + \frac{3 a + 2}{4 ( a + 1 ) ^{\frac{3}{2}}}

Solution steps

Compute the tangent line to the general point

x = a

Find the tangent point:

(a, \frac{1}{2 a + 1})

Find the slope of

y = \frac{1}{2 x + 1} : \frac{d y}{d x} = - \frac{1}{4 ( x + 1 ) ^{\frac{3}{2}}}

Compute the slope of

y = \frac{1}{2 x + 1}

x = a : m = - \frac{1}{4 ( a + 1 ) ^{\frac{3}{2}}}

Find the line with slope m=

- \frac{1}{4 ( a + 1 ) ^{\frac{3}{2}}}

and passing through

(a, \frac{1}{2 a + 1}) : y = - \frac{1}{4 ( a + 1 ) ^{\frac{3}{2}}} x + \frac{3 a + 2}{4 ( a + 1 ) ^{\frac{3}{2}}}

y = - \frac{1}{4 ( a + 1 ) ^{\frac{3}{2}}} x + \frac{3 a + 2}{4 ( a + 1 ) ^{\frac{3}{2}}}

\frac{d y}{d x} = \frac{9}{x}

\frac{x ^{2} - 3 x + 2}{2 x + 1}

\frac{e ^{- 2 s}}{s ^{2} - 2 s + 5}

\frac{\partial}{\partial x} ((7 x + 3 y)^{5})

\frac{\partial}{\partial y} ((- x^{4} + 7 y^{2} + 3 y)^{6})

What is the tangent of 1/(2sqrt(x+1)),\at a=3 ?
The tangent of 1/(2sqrt(x+1)),\at a=3 is y=-1/(4(a+1)^{3/2)}x+(3a+2)/(4(a+1)^{3/2)}