What is the derivative of ln((x^2(x+1))/(sqrt(x+2))) ?

The derivative of ln((x^2(x+1))/(sqrt(x+2))) is (5x^2+15x+8)/(2x(x+1)(x+2))

What is the first derivative of ln((x^2(x+1))/(sqrt(x+2))) ?

The first derivative of ln((x^2(x+1))/(sqrt(x+2))) is (5x^2+15x+8)/(2x(x+1)(x+2))

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derivative of

ln (\frac{x ^{2} ( x + 1 )}{x + 2})

\frac{5 x ^{2} + 15 x + 8}{2 x ( x + 1 ) ( x + 2 )}

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (\frac{x ^{2} ( x + 1 )}{x + 2}) = \frac{5 x ^{3} + 15 x ^{2} + 8 x}{2 ( x + 2 ) x + 2}

= \frac{1}{\frac{x ^{2} ( x + 1 )}{x + 2}} \cdot \frac{5 x ^{3} + 15 x ^{2} + 8 x}{2 ( x + 2 ) x + 2}

Simplify

\frac{1}{\frac{x ^{2} ( x + 1 )}{x + 2}} \cdot \frac{5 x ^{3} + 15 x ^{2} + 8 x}{2 ( x + 2 ) x + 2} : \frac{5 x ^{2} + 15 x + 8}{2 x ( x + 1 ) ( x + 2 )}

= \frac{5 x ^{2} + 15 x + 8}{2 x ( x + 1 ) ( x + 2 )}

\int \frac{5 + 4 x}{1 + x ^{2}} d x

\int e^{- \frac{x}{8}} d x

x^{^{''}} + 2 x^{^{'}} + 2 = 0, x (0) = 0, x^{^{'}} (0) = 1

x \to 0 lim (\frac{1}{x ^{3} - 7 x})

f (x) = 2 x^{3} + 3 x^{2} - 5 x, - 4, 3

What is the derivative of ln((x^2(x+1))/(sqrt(x+2))) ?
The derivative of ln((x^2(x+1))/(sqrt(x+2))) is (5x^2+15x+8)/(2x(x+1)(x+2))
What is the first derivative of ln((x^2(x+1))/(sqrt(x+2))) ?
The first derivative of ln((x^2(x+1))/(sqrt(x+2))) is (5x^2+15x+8)/(2x(x+1)(x+2))