What is the derivative of xsin(xy) ?

The derivative of xsin(xy) is sin(xy)+xcos(xy)(y+x(dy)/(dx))

What is the first derivative of xsin(xy) ?

The first derivative of xsin(xy) is sin(xy)+xcos(xy)(y+x(dy)/(dx))

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

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

Solution steps

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

Apply the Product Rule:

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

f = x, g = sin (x y)

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

\frac{d x}{d x} = 1

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

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

Simplify

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

What is the derivative of xsin(xy) ?
The derivative of xsin(xy) is sin(xy)+xcos(xy)(y+x(dy)/(dx))
What is the first derivative of xsin(xy) ?
The first derivative of xsin(xy) is sin(xy)+xcos(xy)(y+x(dy)/(dx))