What is the tangent of f(x)=(3+4x^{1/2})/(2+x^{3/2)} ?

The tangent of f(x)=(3+4x^{1/2})/(2+x^{3/2)} is

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tangent of f(x)=(3+4x^{1/2})/(2+x^{3/2)}

tangent of

f (x) = \frac{3 + 4 x ^{\frac{1}{2}}}{2 + x ^{\frac{3}{2}}}

y = \frac{- 9 a _{0} - 8 a _{0}^{\frac{3}{2}} + 8}{2 a _{0}^{\frac{1}{2}} ( 2 + a _{0}^{\frac{3}{2}} ) ^{2}} x + \frac{15 a _{0}^{3} + 40 a _{0}^{2} + 16 a _{0}^{\frac{7}{2}} + 42 a _{0}^{\frac{3}{2}} + 16 a _{0}^{\frac{1}{2}} + 24}{2 ( a _{0}^{\frac{3}{2}} + 2 ) ^{3}}

Solution steps

Compute the tangent line to the general point

x = a_{0}

Find the tangent point:

(a_{0}, \frac{3 + 4 a _{0}^{\frac{1}{2}}}{2 + a _{0}^{\frac{3}{2}}})

Find the slope of

f (x) = \frac{3 + 4 x ^{\frac{1}{2}}}{2 + x ^{\frac{3}{2}}} : \frac{df ( x )}{d x} = \frac{- 9 x - 8 x ^{\frac{3}{2}} + 8}{2 x ^{\frac{1}{2}} ( 2 + x ^{\frac{3}{2}} ) ^{2}}

Compute the slope of

f (x) = \frac{3 + 4 x ^{\frac{1}{2}}}{2 + x ^{\frac{3}{2}}}

x = a_{0} : m = \frac{- 9 a _{0} - 8 a _{0}^{\frac{3}{2}} + 8}{2 a _{0}^{\frac{1}{2}} ( 2 + a _{0}^{\frac{3}{2}} ) ^{2}}

Find the line with slope m=

\frac{- 9 a _{0} - 8 a _{0}^{\frac{3}{2}} + 8}{2 a _{0}^{\frac{1}{2}} ( 2 + a _{0}^{\frac{3}{2}} ) ^{2}}

and passing through

(a_{0}, \frac{3 + 4 a _{0}^{\frac{1}{2}}}{2 + a _{0}^{\frac{3}{2}}}) : y = \frac{- 9 a _{0} - 8 a _{0}^{\frac{3}{2}} + 8}{2 a _{0}^{\frac{1}{2}} ( 2 + a _{0}^{\frac{3}{2}} ) ^{2}} x + \frac{15 a _{0}^{3} + 40 a _{0}^{2} + 16 a _{0}^{\frac{7}{2}} + 42 a _{0}^{\frac{3}{2}} + 16 a _{0}^{\frac{1}{2}} + 24}{2 ( a _{0}^{\frac{3}{2}} + 2 ) ^{3}}

y = \frac{- 9 a _{0} - 8 a _{0}^{\frac{3}{2}} + 8}{2 a _{0}^{\frac{1}{2}} ( 2 + a _{0}^{\frac{3}{2}} ) ^{2}} x + \frac{15 a _{0}^{3} + 40 a _{0}^{2} + 16 a _{0}^{\frac{7}{2}} + 42 a _{0}^{\frac{3}{2}} + 16 a _{0}^{\frac{1}{2}} + 24}{2 ( a _{0}^{\frac{3}{2}} + 2 ) ^{3}}

\int \frac{x ^{2}}{( 5 - x ^{2} ) ^{3}} d x

n = 0 \sum \infty cos (πn)

\frac{d}{d x} (ln (x^{3} - 2 x^{2} + 2))

\int sec^{4} (x) tan^{4} (x) d x

f (x) = (x^{3} + 7 x^{2} + 5)^{5}

What is the tangent of f(x)=(3+4x^{1/2})/(2+x^{3/2)} ?
The tangent of f(x)=(3+4x^{1/2})/(2+x^{3/2)} is