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

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

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

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

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derivative of

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

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

Solution steps

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

Apply the Product Rule:

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

f = x^{\frac{1}{2}}, g = 2 x^{\frac{1}{2}} - 1

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

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

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

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

Simplify

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

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

- \frac{1}{5} x^{2} - \frac{6}{5} x + \frac{7}{5}, [- 5, 5]

y^{^{'}} = 24.5 - \frac{7 y}{( 280 - 2 x )}

\int \frac{1}{x} + 2 d x

\int \frac{x ^{3}}{9 x ^{2} + 7} d x

x e^{x}

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