What is the derivative of x^2y^4e^y ?

The derivative of x^2y^4e^y is 2xy^4e^y+(4y^3(dy)/(dx)e^y+e^y(dy)/(dx)y^4)x^2

What is the first derivative of x^2y^4e^y ?

The first derivative of x^2y^4e^y is 2xy^4e^y+(4y^3(dy)/(dx)e^y+e^y(dy)/(dx)y^4)x^2

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\frac{d}{d x} (x^{2} y^{4} e^{y})

2 x y^{4} e^{y} + (4 y^{3} \frac{d y}{d x} e^{y} + e^{y} \frac{d y}{d x} y^{4}) x^{2}

Solution steps

\frac{d}{d x} (x^{2} y^{4} e^{y})

Apply the Product Rule:

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

f = x^{2}, g = y^{4} e^{y}

= \frac{d}{d x} (x^{2}) y^{4} e^{y} + \frac{d}{d x} (y^{4} e^{y}) x^{2}

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

\frac{d}{d x} (y^{4} e^{y}) = 4 y^{3} \frac{d y}{d x} e^{y} + e^{y} \frac{d y}{d x} y^{4}

= 2 x y^{4} e^{y} + (4 y^{3} \frac{d y}{d x} e^{y} + e^{y} \frac{d y}{d x} y^{4}) x^{2}

\frac{d}{d x} (\frac{x ^{2}}{x ^{2} - 6})

6 - x^{2}, x^{2} - 20 < x < 3

4 x - 20

x \to \infty lim (x^{4} e^{x})

ln (5 - x)

What is the derivative of x^2y^4e^y ?
The derivative of x^2y^4e^y is 2xy^4e^y+(4y^3(dy)/(dx)e^y+e^y(dy)/(dx)y^4)x^2
What is the first derivative of x^2y^4e^y ?
The first derivative of x^2y^4e^y is 2xy^4e^y+(4y^3(dy)/(dx)e^y+e^y(dy)/(dx)y^4)x^2