What is the d/(ds)(ln((s^8x^7e^{-sx^8})^n)) ?

The d/(ds)(ln((s^8x^7e^{-sx^8})^n)) is (n(-sx^8+8))/s

What is the first d/(ds)(ln((s^8x^7e^{-sx^8})^n)) ?

The first d/(ds)(ln((s^8x^7e^{-sx^8})^n)) is (n(-sx^8+8))/s

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\frac{d}{d s} (ln ((s^{8} x^{7} e^{- s x^{8}})^{n}))

\frac{n ( - s x ^{8} + 8 )}{s}

Solution steps

\frac{d}{d s} (ln ((s^{8} x^{7} e^{- s x^{8}})^{n}))

Treat

n, x

as a constant

Apply the chain rule:

\frac{1}{( s ^{8} x ^{7} e ^{- s x^{8}} ) ^{n}} \frac{d}{d s} ((s^{8} x^{7} e^{- s x^{8}})^{n})

= \frac{1}{( s ^{8} x ^{7} e ^{- s x^{8}} ) ^{n}} \frac{d}{d s} ((s^{8} x^{7} e^{- s x^{8}})^{n})

\frac{d}{d s} ((s^{8} x^{7} e^{- s x^{8}})^{n}) = n e^{- x^{8} s (n - 1)} x^{7 n} s^{8 (n - 1)} (8 e^{- x^{8} s} s^{7} - x^{8} e^{- x^{8} s} s^{8})

= \frac{1}{( s ^{8} x ^{7} e ^{- s x^{8}} ) ^{n}} n e^{- x^{8} s (n - 1)} x^{7 n} s^{8 (n - 1)} (8 e^{- x^{8} s} s^{7} - x^{8} e^{- x^{8} s} s^{8})

Simplify

\frac{1}{( s ^{8} x ^{7} e ^{- s x^{8}} ) ^{n}} n e^{- x^{8} s (n - 1)} x^{7 n} s^{8 (n - 1)} (8 e^{- x^{8} s} s^{7} - x^{8} e^{- x^{8} s} s^{8}) : \frac{n ( - s x ^{8} + 8 )}{s}

= \frac{n ( - s x ^{8} + 8 )}{s}

\int 42 x d x

\frac{d}{d x} ((\frac{x ^{2} + 7}{x ^{2} - 7})^{3})

\frac{d y}{d x}, x + y = 5

\frac{\partial}{\partial x} (\frac{x}{2})

What is the d/(ds)(ln((s^8x^7e^{-sx^8})^n)) ?
The d/(ds)(ln((s^8x^7e^{-sx^8})^n)) is (n(-sx^8+8))/s
What is the first d/(ds)(ln((s^8x^7e^{-sx^8})^n)) ?
The first d/(ds)(ln((s^8x^7e^{-sx^8})^n)) is (n(-sx^8+8))/s