What is the derivative of ((5x+1/(2sqrt(x)))dx) ?

The derivative of ((5x+1/(2sqrt(x)))dx) is (d(15x+1))/(4sqrt(x))

What is the first derivative of ((5x+1/(2sqrt(x)))dx) ?

The first derivative of ((5x+1/(2sqrt(x)))dx) is (d(15x+1))/(4sqrt(x))

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\frac{d}{d x} ((\frac{5 x + 1}{2 x}) d x)

\frac{d ( 15 x + 1 )}{4 x}

Solution steps

\frac{d}{d x} ((\frac{5 x + 1}{2 x}) d x)

Treat

d

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{2} d \frac{d}{d x} (\frac{5 x + 1}{x} x)

Apply the Product Rule:

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

f = \frac{5 x + 1}{x}, g = x

= \frac{1}{2} d (\frac{d}{d x} (\frac{5 x + 1}{x}) x + \frac{d x}{d x} \frac{5 x + 1}{x})

\frac{d}{d x} (\frac{5 x + 1}{x}) = \frac{5 x - 1}{2 x x}

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

= \frac{1}{2} d (\frac{5 x - 1}{2 x x} x + 1 \cdot \frac{5 x + 1}{x})

Simplify

\frac{1}{2} d (\frac{5 x - 1}{2 x x} x + 1 \cdot \frac{5 x + 1}{x}) : \frac{d ( 15 x + 1 )}{4 x}

= \frac{d ( 15 x + 1 )}{4 x}

What is the derivative of ((5x+1/(2sqrt(x)))dx) ?
The derivative of ((5x+1/(2sqrt(x)))dx) is (d(15x+1))/(4sqrt(x))
What is the first derivative of ((5x+1/(2sqrt(x)))dx) ?
The first derivative of ((5x+1/(2sqrt(x)))dx) is (d(15x+1))/(4sqrt(x))