What is the derivative of {f}(x(x)^{-1}) ?

The derivative of {f}(x(x)^{-1}) is (derivative of {f}(x))/x-({f}(x))/(x^2)

What is the first derivative of {f}(x(x)^{-1}) ?

The first derivative of {f}(x(x)^{-1}) is (derivative of {f}(x))/x-({f}(x))/(x^2)

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\frac{d}{d x} (f (x) (x)^{- 1})

\frac{\frac{d}{d x} ( f ( x ) )}{x} - \frac{f ( x )}{x ^{2}}

Solution steps

\frac{d}{d x} (f (x) (x)^{- 1})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = f (x), g = x^{- 1}

= \frac{d}{d x} (f (x)) x^{- 1} + \frac{d}{d x} (x^{- 1}) f (x)

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

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

Simplify

\frac{d}{d x} (f (x)) x^{- 1} + (- \frac{1}{x ^{2}}) f (x) : \frac{\frac{d}{d x} ( f ( x ) )}{x} - \frac{f ( x )}{x ^{2}}

= \frac{\frac{d}{d x} ( f ( x ) )}{x} - \frac{f ( x )}{x ^{2}}

What is the derivative of {f}(x(x)^{-1}) ?
The derivative of {f}(x(x)^{-1}) is (derivative of {f}(x))/x-({f}(x))/(x^2)
What is the first derivative of {f}(x(x)^{-1}) ?
The first derivative of {f}(x(x)^{-1}) is (derivative of {f}(x))/x-({f}(x))/(x^2)