What is (\partial)/(\partial t)(te^{st}) ?

The answer to (\partial)/(\partial t)(te^{st}) is e^{ts}+e^{ts}ts

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(\partial)/(\partial t)(te^{st})

\frac{\partial}{\partial t} (t e^{s t})

e^{t s} + e^{t s} t s

Solution steps

\frac{\partial}{\partial t} (t e^{s t})

Treat

s

as a constant

Apply the Product Rule:

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

f = t, g = e^{s t}

= \frac{\partial}{\partial t} (t) e^{s t} + \frac{\partial}{\partial t} (e^{s t}) t

\frac{\partial}{\partial t} (t) = 1

\frac{\partial}{\partial t} (e^{s t}) = e^{s t} s

= 1 \cdot e^{s t} + e^{s t} s t

Simplify

= e^{t s} + e^{t s} t s

What is (\partial)/(\partial t)(te^{st}) ?
The answer to (\partial)/(\partial t)(te^{st}) is e^{ts}+e^{ts}ts