What is the derivative of xe^{xsin(x}) ?

The derivative of xe^{xsin(x}) is e^{xsin(x)}+e^{xsin(x)}x(sin(x)+xcos(x))

What is the first derivative of xe^{xsin(x}) ?

The first derivative of xe^{xsin(x}) is e^{xsin(x)}+e^{xsin(x)}x(sin(x)+xcos(x))

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\frac{d}{d x} (x e^{x s i n (x)})

e^{x s i n (x)} + e^{x s i n (x)} x (sin (x) + x cos (x))

Solution steps

\frac{d}{d x} (x e^{x s i n (x)})

Apply the Product Rule:

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

f = x, g = e^{x s i n (x)}

= \frac{d x}{d x} e^{x s i n (x)} + \frac{d}{d x} (e^{x s i n (x)}) x

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

\frac{d}{d x} (e^{x s i n (x)}) = e^{x s i n (x)} (sin (x) + x cos (x))

= 1 \cdot e^{x s i n (x)} + e^{x s i n (x)} (sin (x) + x cos (x)) x

Simplify

= e^{x s i n (x)} + e^{x s i n (x)} x (sin (x) + x cos (x))

What is the derivative of xe^{xsin(x}) ?
The derivative of xe^{xsin(x}) is e^{xsin(x)}+e^{xsin(x)}x(sin(x)+xcos(x))
What is the first derivative of xe^{xsin(x}) ?
The first derivative of xe^{xsin(x}) is e^{xsin(x)}+e^{xsin(x)}x(sin(x)+xcos(x))