What is the derivative of (8+x{f}(x)/(sqrt(x))) ?

The derivative of (8+x{f}(x)/(sqrt(x))) is (2x^2 derivative of {f}(x)+x{f}(x)-8)/(2xsqrt(x))

What is the first derivative of (8+x{f}(x)/(sqrt(x))) ?

The first derivative of (8+x{f}(x)/(sqrt(x))) is (2x^2 derivative of {f}(x)+x{f}(x)-8)/(2xsqrt(x))

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\frac{d}{d x} (\frac{8 + x f ( x )}{x})

\frac{2 x ^{2} \frac{d}{d x} ( f ( x ) ) + x f ( x ) - 8}{2 x x}

Solution steps

\frac{d}{d x} (\frac{8 + x f ( x )}{x})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 8 + x f ( x ) ) x - \frac{d}{d x} ( x ) ( 8 + x f ( x ) )}{( x ) ^{2}}

\frac{d}{d x} (8 + x f (x)) = x \frac{d}{d x} (f (x)) + f (x)

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

= \frac{( x \frac{d}{d x} ( f ( x ) ) + f ( x ) ) x - \frac{1}{2 x} ( 8 + x f ( x ) )}{( x ) ^{2}}

Simplify

\frac{( x \frac{d}{d x} ( f ( x ) ) + f ( x ) ) x - \frac{1}{2 x} ( 8 + x f ( x ) )}{( x ) ^{2}} : \frac{2 x ^{2} \frac{d}{d x} ( f ( x ) ) + x f ( x ) - 8}{2 x x}

= \frac{2 x ^{2} \frac{d}{d x} ( f ( x ) ) + x f ( x ) - 8}{2 x x}

What is the derivative of (8+x{f}(x)/(sqrt(x))) ?
The derivative of (8+x{f}(x)/(sqrt(x))) is (2x^2 derivative of {f}(x)+x{f}(x)-8)/(2xsqrt(x))
What is the first derivative of (8+x{f}(x)/(sqrt(x))) ?
The first derivative of (8+x{f}(x)/(sqrt(x))) is (2x^2 derivative of {f}(x)+x{f}(x)-8)/(2xsqrt(x))