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

The derivative of ln(a+xsqrt(2ax+x^2)) is (2x^2+3ax)/((a+xsqrt(2ax+x^2))\sqrt{x^2+2ax)}

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

The first derivative of ln(a+xsqrt(2ax+x^2)) is (2x^2+3ax)/((a+xsqrt(2ax+x^2))\sqrt{x^2+2ax)}

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

\frac{2 x ^{2} + 3 a x}{( a + x 2 a x + x ^{2} ) x ^{2} + 2 a x}

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{a + x 2 a x + x ^{2}} \cdot \frac{2 x ^{2} + 3 a x}{x ^{2} + 2 a x}

Simplify

\frac{1}{a + x 2 a x + x ^{2}} \cdot \frac{2 x ^{2} + 3 a x}{x ^{2} + 2 a x} : \frac{2 x ^{2} + 3 a x}{( a + x 2 a x + x ^{2} ) x ^{2} + 2 a x}

= \frac{2 x ^{2} + 3 a x}{( a + x 2 a x + x ^{2} ) x ^{2} + 2 a x}

x \to 3 lim (f (x) x)

\frac{d}{d t} (2 t + 4)

\frac{1}{z ^{2}}

x \to - 5 lim (3 x - 1)

\frac{d}{d x} ((4 x^{2} + 3) (2 x - 5))

What is the derivative of ln(a+xsqrt(2ax+x^2)) ?
The derivative of ln(a+xsqrt(2ax+x^2)) is (2x^2+3ax)/((a+xsqrt(2ax+x^2))\sqrt{x^2+2ax)}
What is the first derivative of ln(a+xsqrt(2ax+x^2)) ?
The first derivative of ln(a+xsqrt(2ax+x^2)) is (2x^2+3ax)/((a+xsqrt(2ax+x^2))\sqrt{x^2+2ax)}