What is (\partial)/(\partial x)(2(x-y)e^{2y+5x^2}) ?

The answer to (\partial)/(\partial x)(2(x-y)e^{2y+5x^2}) is 2(e^{2y+5x^2}+10e^{2y+5x^2}x(x-y))

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(\partial)/(\partial x)(2(x-y)e^{2y+5x^2})

\frac{\partial}{\partial x} (2 (x - y) e^{2 y + 5 x^{2}})

2 (e^{2 y + 5 x^{2}} + 10 e^{2 y + 5 x^{2}} x (x - y))

Solution steps

\frac{\partial}{\partial x} (2 (x - y) e^{2 y + 5 x^{2}})

Treat

y

as a constant

Take the constant out:

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

= 2 \frac{\partial}{\partial x} ((x - y) e^{2 y + 5 x^{2}})

Apply the Product Rule:

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

f = x - y, g = e^{2 y + 5 x^{2}}

= 2 (\frac{\partial}{\partial x} (x - y) e^{2 y + 5 x^{2}} + \frac{\partial}{\partial x} (e^{2 y + 5 x^{2}}) (x - y))

\frac{\partial}{\partial x} (x - y) = 1

\frac{\partial}{\partial x} (e^{2 y + 5 x^{2}}) = e^{2 y + 5 x^{2}} \cdot 10 x

= 2 (1 \cdot e^{2 y + 5 x^{2}} + e^{2 y + 5 x^{2}} \cdot 10 x (x - y))

Simplify

= 2 (e^{2 y + 5 x^{2}} + 10 e^{2 y + 5 x^{2}} x (x - y))

f^{^{'}} (t) - f (t) = - 7 t, f (2) = - 4

\int \frac{2 y}{x ^{2} + y ^{2}} d x

\int e^{s i n (π x)} cos (π x) d x

\int_{0}^{3} \frac{- 6}{x x} d x

(1 - 5 y + x) y^{^{'}} + y = 5 x - 1

What is (\partial)/(\partial x)(2(x-y)e^{2y+5x^2}) ?
The answer to (\partial)/(\partial x)(2(x-y)e^{2y+5x^2}) is 2(e^{2y+5x^2}+10e^{2y+5x^2}x(x-y))