What is the derivative of ln(a+x+sqrt(x^2+2ax)) ?

The derivative of ln(a+x+sqrt(x^2+2ax)) is 1/(sqrt(x^2+2ax))

What is the first derivative of ln(a+x+sqrt(x^2+2ax)) ?

The first derivative of ln(a+x+sqrt(x^2+2ax)) is 1/(sqrt(x^2+2ax))

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

\frac{1}{x ^{2} + 2 a x}

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{a + x + x ^{2} + 2 a x} (\frac{x + a}{x ^{2} + 2 a x} + 1) : \frac{1}{x ^{2} + 2 a x}

= \frac{1}{x ^{2} + 2 a x}

\frac{\partial}{\partial x} (- 2 x y z^{2})

\int_{0}^{1} \frac{1}{x ^{1}} d x

\int \frac{x ^{3} + 8}{( x ^{2} - 1 ) ( x - 2 )} d x

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

y = 2 x, x = 3

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