What is (\partial)/(\partial x)(3x^4e^{x^2y}) ?

The answer to (\partial)/(\partial x)(3x^4e^{x^2y}) is 3(4e^{x^2y}x^3+2e^{x^2y}x^5y)

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(\partial)/(\partial x)(3x^4e^{x^2y})

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

3 (4 e^{x^{2} y} x^{3} + 2 e^{x^{2} y} x^{5} y)

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x^{4}) = 4 x^{3}

\frac{\partial}{\partial x} (e^{x^{2} y}) = e^{x^{2} y} \cdot 2 y x

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

e^{x^{2} y} \cdot 2 y x x^{4} = 2 e^{x^{2} y} x^{5} y

= 3 (4 e^{x^{2} y} x^{3} + 2 e^{x^{2} y} x^{5} y)

x \to - 3 + lim (\frac{3}{x ^{2} - 9})

x \frac{d y}{d x} + 22 y = \frac{1}{x}

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

y^{^{'}} (x) = \frac{6 x ^{2}}{6 y ^{2} - 7}, y (1) = 0

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

What is (\partial)/(\partial x)(3x^4e^{x^2y}) ?
The answer to (\partial)/(\partial x)(3x^4e^{x^2y}) is 3(4e^{x^2y}x^3+2e^{x^2y}x^5y)