What is the derivative of e^xcos(5x) ?

The derivative of e^xcos(5x) is e^xcos(5x)-5e^xsin(5x)

What is the first derivative of e^xcos(5x) ?

The first derivative of e^xcos(5x) is e^xcos(5x)-5e^xsin(5x)

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\frac{d}{d x} (e^{x} cos (5 x))

e^{x} cos (5 x) - 5 e^{x} sin (5 x)

Solution steps

\frac{d}{d x} (e^{x} cos (5 x))

Apply the Product Rule:

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

f = e^{x}, g = cos (5 x)

= \frac{d}{d x} (e^{x}) cos (5 x) + \frac{d}{d x} (cos (5 x)) e^{x}

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

\frac{d}{d x} (cos (5 x)) = - sin (5 x) \cdot 5

= e^{x} cos (5 x) + (- sin (5 x) \cdot 5) e^{x}

Simplify

= e^{x} cos (5 x) - 5 e^{x} sin (5 x)

What is the derivative of e^xcos(5x) ?
The derivative of e^xcos(5x) is e^xcos(5x)-5e^xsin(5x)
What is the first derivative of e^xcos(5x) ?
The first derivative of e^xcos(5x) is e^xcos(5x)-5e^xsin(5x)