What is the derivative of e^θsin(5θ) ?

The derivative of e^θsin(5θ) is e^θsin(5θ)+cos(5θ)*5e^θ

What is the first derivative of e^θsin(5θ) ?

The first derivative of e^θsin(5θ) is e^θsin(5θ)+cos(5θ)*5e^θ

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derivative

e^{θ} sin (5 θ) + cos (5 θ) \cdot 5 e^{θ}

Solution steps

\frac{d}{d θ} (e^{θ} sin (5 θ))

Apply the Product Rule:

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

f = e^{θ}, g = sin (5 θ)

= \frac{d}{d θ} (e^{θ}) sin (5 θ) + \frac{d}{d θ} (sin (5 θ)) e^{θ}

\frac{d}{d θ} (e^{θ}) = e^{θ}

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

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

What is the derivative of e^θsin(5θ) ?
The derivative of e^θsin(5θ) is e^θsin(5θ)+cos(5θ)*5e^θ
What is the first derivative of e^θsin(5θ) ?
The first derivative of e^θsin(5θ) is e^θsin(5θ)+cos(5θ)*5e^θ