What is the d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) ?

The d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) is -(2d^2z)/((d^2+z^2)(d^2-z^2))

What is the first d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) ?

The first d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) is -(2d^2z)/((d^2+z^2)(d^2-z^2))

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\frac{d}{d z} (ln (\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}}))

- \frac{2 d ^{2} z}{( d ^{2} + z ^{2} ) ( d ^{2} - z ^{2} )}

Solution steps

\frac{d}{d z} (ln (\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}}))

Treat

d

as a constant

Apply the chain rule:

\frac{1}{\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}}} \frac{d}{d z} (\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}})

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

\frac{d}{d z} (\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}}) = - \frac{2 d ^{2} z}{- z ^{2} + d ^{2} ( z ^{2} + d ^{2} ) ^{\frac{3}{2}}}

= \frac{1}{\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}}} (- \frac{2 d ^{2} z}{- z ^{2} + d ^{2} ( z ^{2} + d ^{2} ) ^{\frac{3}{2}}})

Simplify

\frac{1}{\frac{d ^{2} - z ^{2}}{d ^{2} + z ^{2}}} (- \frac{2 d ^{2} z}{- z ^{2} + d ^{2} ( z ^{2} + d ^{2} ) ^{\frac{3}{2}}}) : - \frac{2 d ^{2} z}{( d ^{2} + z ^{2} ) ( d ^{2} - z ^{2} )}

= - \frac{2 d ^{2} z}{( d ^{2} + z ^{2} ) ( d ^{2} - z ^{2} )}

\frac{x}{9 - x ^{2}}

x \to - 3 lim (- 1.2 x - 4.6)

\frac{d}{d x} (\frac{x ^{2}}{1 + 6 x})

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

\frac{d}{d x} (- 4 tan (x) - csc (x))

What is the d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) ?
The d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) is -(2d^2z)/((d^2+z^2)(d^2-z^2))
What is the first d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) ?
The first d/(dz)(ln(sqrt((d^2-z^2)/(d^2+z^2)))) is -(2d^2z)/((d^2+z^2)(d^2-z^2))