What is the derivative of e^ysin(xy) ?

The derivative of e^ysin(xy) is e^y(dy)/(dx)sin(xy)+cos(xy)(y+x(dy)/(dx))e^y

What is the first derivative of e^ysin(xy) ?

The first derivative of e^ysin(xy) is e^y(dy)/(dx)sin(xy)+cos(xy)(y+x(dy)/(dx))e^y

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\frac{d}{d x} (e^{y} sin (x y))

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

Solution steps

\frac{d}{d x} (e^{y} sin (x y))

Apply the Product Rule:

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

f = e^{y}, g = sin (x y)

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

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

\frac{d}{d x} (sin (x y)) = cos (x y) (y + x \frac{d y}{d x})

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

x \to - \infty lim (2 x - x^{2})

\int 2 x x^{2} - 1 d x

n = 0 \sum \infty (- 0.6)^{n}

\frac{0.3 ( 5 ^{x} )}{x ^{3}}

cos (x^{2}), 0

What is the derivative of e^ysin(xy) ?
The derivative of e^ysin(xy) is e^y(dy)/(dx)sin(xy)+cos(xy)(y+x(dy)/(dx))e^y
What is the first derivative of e^ysin(xy) ?
The first derivative of e^ysin(xy) is e^y(dy)/(dx)sin(xy)+cos(xy)(y+x(dy)/(dx))e^y