What is the derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) ?

The derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) is (4x^3+11x^2+20x+21)/(2(x^2+3)(x+2))

What is the first derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) ?

The first derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) is (4x^3+11x^2+20x+21)/(2(x^2+3)(x+2))

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derivative of

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

\frac{4 x ^{3} + 11 x ^{2} + 20 x + 21}{2 ( x ^{2} + 3 ) ( x + 2 )}

Solution steps

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

Apply the chain rule:

\frac{1}{\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2}} \frac{d}{d x} (\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2})

= \frac{1}{\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2}} \frac{d}{d x} (\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2})

\frac{d}{d x} (\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2}) = \frac{4 e ^{2 x} x ^{3} + 11 e ^{2 x} x ^{2} + 20 e ^{2 x} x + 21 e ^{2 x}}{2 ( x + 2 ) x + 2}

= \frac{1}{\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2}} \cdot \frac{4 e ^{2 x} x ^{3} + 11 e ^{2 x} x ^{2} + 20 e ^{2 x} x + 21 e ^{2 x}}{2 ( x + 2 ) x + 2}

Simplify

\frac{1}{\frac{e ^{2 x} ( x ^{2} + 3 )}{x + 2}} \cdot \frac{4 e ^{2 x} x ^{3} + 11 e ^{2 x} x ^{2} + 20 e ^{2 x} x + 21 e ^{2 x}}{2 ( x + 2 ) x + 2} : \frac{4 x ^{3} + 11 x ^{2} + 20 x + 21}{2 ( x ^{2} + 3 ) ( x + 2 )}

= \frac{4 x ^{3} + 11 x ^{2} + 20 x + 21}{2 ( x ^{2} + 3 ) ( x + 2 )}

- x^{2} + 3 x - 2

\frac{d}{d x} (x e^{- b x})

\int (5 t^{2} + 3 t) d t

\frac{d}{d x} (sec (x))

6 x + 7, x, 0, 8

What is the derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) ?
The derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) is (4x^3+11x^2+20x+21)/(2(x^2+3)(x+2))
What is the first derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) ?
The first derivative of ln((e^{2x}(x^2+3))/(sqrt(x+2))) is (4x^3+11x^2+20x+21)/(2(x^2+3)(x+2))