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

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

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

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

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

f (x) + x \frac{d}{d x} (f (x))

Solution steps

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

Simplify

f (x) (x)^{- 1} (x^{2}) : x f (x)

= \frac{d}{d x} (x f (x))

Apply the Product Rule:

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

f = x, g = f (x)

= \frac{d x}{d x} f (x) + \frac{d}{d x} (f (x)) x

\frac{d x}{d x} = 1

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

Simplify

= f (x) + x \frac{d}{d x} (f (x))

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