What is the derivative of f(x)=(x+x^3)/(sqrt(x)) ?

The derivative of f(x)=(x+x^3)/(sqrt(x)) is (1+5x^2)/(2sqrt(x))

What is the first derivative of f(x)=(x+x^3)/(sqrt(x)) ?

The first derivative of f(x)=(x+x^3)/(sqrt(x)) is (1+5x^2)/(2sqrt(x))

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derivative of

f (x) = \frac{x + x ^{3}}{x}

\frac{1 + 5 x ^{2}}{2 x}

Solution steps

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

Simplify

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

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

Apply the Product Rule:

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

f = x, g = 1 + x^{2}

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

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

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

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

Simplify

\frac{1}{2 x} (1 + x^{2}) + 2 x x : \frac{1 + 5 x ^{2}}{2 x}

= \frac{1 + 5 x ^{2}}{2 x}

What is the derivative of f(x)=(x+x^3)/(sqrt(x)) ?
The derivative of f(x)=(x+x^3)/(sqrt(x)) is (1+5x^2)/(2sqrt(x))
What is the first derivative of f(x)=(x+x^3)/(sqrt(x)) ?
The first derivative of f(x)=(x+x^3)/(sqrt(x)) is (1+5x^2)/(2sqrt(x))