What is the derivative of x^{0.5}y^{0.5} ?

The derivative of x^{0.5}y^{0.5} is (0.5y^{1/2})/(x^{1/2)}+(0.5x^{1/2}\frac{dy)/(dx)}{y^{1/2}}

What is the first derivative of x^{0.5}y^{0.5} ?

The first derivative of x^{0.5}y^{0.5} is (0.5y^{1/2})/(x^{1/2)}+(0.5x^{1/2}\frac{dy)/(dx)}{y^{1/2}}

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\frac{d}{d x} (x^{0.5} y^{0.5})

\frac{0.5 y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}} + \frac{0.5 x ^{\frac{1}{2}} \frac{d y}{d x}}{y ^{\frac{1}{2}}}

Solution steps

\frac{d}{d x} (x^{0.5} y^{0.5})

Apply the Product Rule:

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

f = x^{0.5}, g = y^{0.5}

= \frac{d}{d x} (x^{0.5}) y^{0.5} + \frac{d}{d x} (y^{0.5}) x^{0.5}

\frac{d}{d x} (x^{0.5}) = \frac{0.5}{x ^{0.5}}

\frac{d}{d x} (y^{0.5}) = \frac{0.5 \frac{d y}{d x}}{y ^{\frac{1}{2}}}

= \frac{0.5}{x ^{0.5}} y^{0.5} + \frac{0.5 \frac{d y}{d x}}{y ^{\frac{1}{2}}} x^{0.5}

Simplify

\frac{0.5}{x ^{\frac{1}{2}}} y^{\frac{1}{2}} + \frac{0.5 \frac{d y}{d x}}{y ^{\frac{1}{2}}} x^{\frac{1}{2}} : \frac{0.5 y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}} + \frac{0.5 x ^{\frac{1}{2}} \frac{d y}{d x}}{y ^{\frac{1}{2}}}

= \frac{0.5 y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}} + \frac{0.5 x ^{\frac{1}{2}} \frac{d y}{d x}}{y ^{\frac{1}{2}}}

\frac{\partial}{\partial x} (3 e^{3 x + 4 y} cos (5 z))

(6 + x) y^{^{'}} = 5 y

\int 16 x ln (3 x) d x

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

x = - 2, x = 1, y = 2 x^{2} + 12, y = 0

What is the derivative of x^{0.5}y^{0.5} ?
The derivative of x^{0.5}y^{0.5} is (0.5y^{1/2})/(x^{1/2)}+(0.5x^{1/2}\frac{dy)/(dx)}{y^{1/2}}
What is the first derivative of x^{0.5}y^{0.5} ?
The first derivative of x^{0.5}y^{0.5} is (0.5y^{1/2})/(x^{1/2)}+(0.5x^{1/2}\frac{dy)/(dx)}{y^{1/2}}