What is the derivative of e^xsin(e^x) ?

The derivative of e^xsin(e^x) is e^xsin(e^x)+e^{2x}cos(e^x)

What is the first derivative of e^xsin(e^x) ?

The first derivative of e^xsin(e^x) is e^xsin(e^x)+e^{2x}cos(e^x)

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

e^{x} sin (e^{x}) + e^{2 x} cos (e^{x})

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

= e^{x} sin (e^{x}) + cos (e^{x}) e^{x} e^{x}

cos (e^{x}) e^{x} e^{x} = e^{2 x} cos (e^{x})

= e^{x} sin (e^{x}) + e^{2 x} cos (e^{x})

What is the derivative of e^xsin(e^x) ?
The derivative of e^xsin(e^x) is e^xsin(e^x)+e^{2x}cos(e^x)
What is the first derivative of e^xsin(e^x) ?
The first derivative of e^xsin(e^x) is e^xsin(e^x)+e^{2x}cos(e^x)