What is the derivative of (1+xye^{xy}) ?

The derivative of (1+xye^{xy}) is e^{xy}x^2y(dy)/(dx)+2e^{xy}x(dy)/(dx)+e^{xy}xy^2+2e^{xy}y

What is the first derivative of (1+xye^{xy}) ?

The first derivative of (1+xye^{xy}) is e^{xy}x^2y(dy)/(dx)+2e^{xy}x(dy)/(dx)+e^{xy}xy^2+2e^{xy}y

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\frac{d}{d x} ((1 + x y) e^{x y})

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

Solution steps

\frac{d}{d x} ((1 + x y) e^{x y})

Apply the Product Rule:

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

f = 1 + x y, g = e^{x y}

= \frac{d}{d x} (1 + x y) e^{x y} + \frac{d}{d x} (e^{x y}) (1 + x y)

\frac{d}{d x} (1 + x y) = x \frac{d y}{d x} + y

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

= (x \frac{d y}{d x} + y) e^{x y} + e^{x y} (y + x \frac{d y}{d x}) (1 + x y)

Simplify

(x \frac{d y}{d x} + y) e^{x y} + e^{x y} (y + x \frac{d y}{d x}) (1 + x y) : e^{x y} x^{2} y \frac{d y}{d x} + 2 e^{x y} x \frac{d y}{d x} + e^{x y} x y^{2} + 2 e^{x y} y

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

(3 x + 28)^{\frac{1}{2}}

\int e^{\frac{y}{x}} d y

f (x) = 9 - ln (x), at x = 1

\int \frac{1}{x 25 x ^{2} + 1} d x

f (x) = x^{- 3}

What is the derivative of (1+xye^{xy}) ?
The derivative of (1+xye^{xy}) is e^{xy}x^2y(dy)/(dx)+2e^{xy}x(dy)/(dx)+e^{xy}xy^2+2e^{xy}y
What is the first derivative of (1+xye^{xy}) ?
The first derivative of (1+xye^{xy}) is e^{xy}x^2y(dy)/(dx)+2e^{xy}x(dy)/(dx)+e^{xy}xy^2+2e^{xy}y