What is the derivative of ln(h+1/(2ax^2+b)) ?

The derivative of ln(h+1/(2ax^2+b)) is -2/(x(2ahx^2+2abx^2+1))

What is the first derivative of ln(h+1/(2ax^2+b)) ?

The first derivative of ln(h+1/(2ax^2+b)) is -2/(x(2ahx^2+2abx^2+1))

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

- \frac{2}{x ( 2 ah x ^{2} + 2 ab x ^{2} + 1 )}

Solution steps

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

Treat

a, b, h

as a constant

Apply the chain rule:

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

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

\frac{d}{d x} (h + \frac{1}{2 a x ^{2}} + b) = - \frac{1}{a x ^{3}}

= \frac{1}{h + \frac{1}{2 a x ^{2}} + b} (- \frac{1}{a x ^{3}})

Simplify

\frac{1}{h + \frac{1}{2 a x ^{2}} + b} (- \frac{1}{a x ^{3}}) : - \frac{2}{x ( 2 ah x ^{2} + 2 ab x ^{2} + 1 )}

= - \frac{2}{x ( 2 ah x ^{2} + 2 ab x ^{2} + 1 )}

What is the derivative of ln(h+1/(2ax^2+b)) ?
The derivative of ln(h+1/(2ax^2+b)) is -2/(x(2ahx^2+2abx^2+1))
What is the first derivative of ln(h+1/(2ax^2+b)) ?
The first derivative of ln(h+1/(2ax^2+b)) is -2/(x(2ahx^2+2abx^2+1))