What is (\partial)/(\partial t)(4e^{sin(x+3ct)}) ?

The answer to (\partial)/(\partial t)(4e^{sin(x+3ct)}) is 12ce^{sin(x+3ct)}cos(x+3ct)

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(\partial)/(\partial t)(4e^{sin(x+3ct)})

\frac{\partial}{\partial t} (4 e^{s i n (x + 3 c t)})

12 c e^{s i n (x + 3 c t)} cos (x + 3 c t)

Solution steps

\frac{\partial}{\partial t} (4 e^{s i n (x + 3 c t)})

Treat

c, x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{\partial}{\partial t} (e^{s i n (x + 3 c t)})

Apply the chain rule:

e^{s i n (x + 3 c t)} \frac{\partial}{\partial t} (sin (x + 3 c t))

= e^{s i n (x + 3 c t)} \frac{\partial}{\partial t} (sin (x + 3 c t))

\frac{\partial}{\partial t} (sin (x + 3 c t)) = cos (x + 3 c t) \cdot 3 c

= 4 e^{s i n (x + 3 c t)} cos (x + 3 c t) \cdot 3 c

Simplify

= 12 c e^{s i n (x + 3 c t)} cos (x + 3 c t)

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What is (\partial)/(\partial t)(4e^{sin(x+3ct)}) ?
The answer to (\partial)/(\partial t)(4e^{sin(x+3ct)}) is 12ce^{sin(x+3ct)}cos(x+3ct)