What is (\partial)/(\partial x)(1/(1/x+c/x+k)) ?

The answer to (\partial)/(\partial x)(1/(1/x+c/x+k)) is (c+1)/((1+c+kx)^2)

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(\partial)/(\partial x)(1/(1/x+c/x+k))

\frac{\partial}{\partial x} (\frac{1}{\frac{1}{x} + \frac{c}{x} + k})

\frac{c + 1}{( 1 + c + k x ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{1}{\frac{1}{x} + \frac{c}{x} + k})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{\partial}{\partial x} ((\frac{1}{x} + \frac{c}{x} + k)^{- 1})

Apply the chain rule:

- \frac{1}{( \frac{1}{x} + \frac{c}{x} + k ) ^{2}} \frac{\partial}{\partial x} (\frac{1}{x} + \frac{c}{x} + k)

= - \frac{1}{( \frac{1}{x} + \frac{c}{x} + k ) ^{2}} \frac{\partial}{\partial x} (\frac{1}{x} + \frac{c}{x} + k)

\frac{\partial}{\partial x} (\frac{1}{x} + \frac{c}{x} + k) = - \frac{c + 1}{x ^{2}}

= - \frac{1}{( \frac{1}{x} + \frac{c}{x} + k ) ^{2}} (- \frac{c + 1}{x ^{2}})

Simplify

- \frac{1}{( \frac{1}{x} + \frac{c}{x} + k ) ^{2}} (- \frac{c + 1}{x ^{2}}) : \frac{c + 1}{( 1 + c + k x ) ^{2}}

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

\frac{\partial}{\partial x} (- y (y + sin (x)))

1 - \frac{3 x}{x ^{3} + 27}

\int \frac{1}{x ^{2} + 12 x + 35} d x

(- 4.4) (- 1.5)

\frac{d}{d x} (\frac{9}{sin ( x ) + cos ( x )})

What is (\partial)/(\partial x)(1/(1/x+c/x+k)) ?
The answer to (\partial)/(\partial x)(1/(1/x+c/x+k)) is (c+1)/((1+c+kx)^2)