What is the derivative of x^{0.4}y^{0.6} ?

The derivative of x^{0.4}y^{0.6} is (0.4y^{0.6})/(x^{0.6)}+(0.6x^{0.4}\frac{dy)/(dx)}{y^{0.4}}

What is the first derivative of x^{0.4}y^{0.6} ?

The first derivative of x^{0.4}y^{0.6} is (0.4y^{0.6})/(x^{0.6)}+(0.6x^{0.4}\frac{dy)/(dx)}{y^{0.4}}

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\frac{d}{d x} (x^{0.4} y^{0.6})

\frac{0.4 y ^{0.6}}{x ^{0.6}} + \frac{0.6 x ^{0.4} \frac{d y}{d x}}{y ^{0.4}}

Solution steps

\frac{d}{d x} (x^{0.4} y^{0.6})

Apply the Product Rule:

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

f = x^{0.4}, g = y^{0.6}

= \frac{d}{d x} (x^{0.4}) y^{0.6} + \frac{d}{d x} (y^{0.6}) x^{0.4}

\frac{d}{d x} (x^{0.4}) = \frac{0.4}{x ^{0.6}}

\frac{d}{d x} (y^{0.6}) = \frac{0.6 \frac{d y}{d x}}{y ^{0.4}}

= \frac{0.4}{x ^{0.6}} y^{0.6} + \frac{0.6 \frac{d y}{d x}}{y ^{0.4}} x^{0.4}

Simplify

\frac{0.4}{x ^{0.6}} y^{0.6} + \frac{0.6 \frac{d y}{d x}}{y ^{0.4}} x^{0.4} : \frac{0.4 y ^{0.6}}{x ^{0.6}} + \frac{0.6 x ^{0.4} \frac{d y}{d x}}{y ^{0.4}}

= \frac{0.4 y ^{0.6}}{x ^{0.6}} + \frac{0.6 x ^{0.4} \frac{d y}{d x}}{y ^{0.4}}

What is the derivative of x^{0.4}y^{0.6} ?
The derivative of x^{0.4}y^{0.6} is (0.4y^{0.6})/(x^{0.6)}+(0.6x^{0.4}\frac{dy)/(dx)}{y^{0.4}}
What is the first derivative of x^{0.4}y^{0.6} ?
The first derivative of x^{0.4}y^{0.6} is (0.4y^{0.6})/(x^{0.6)}+(0.6x^{0.4}\frac{dy)/(dx)}{y^{0.4}}