Question

Let's derive this exponential function!

e^ (8t sin 2t)

I know the answer is e^ (8t sin 2t) * (8t sin 2t)'

I would love to go step by step to understand how to derive the exponent.

Thank you!

Answer 1

You mean differentiate?

By a general rule:

The differentiation of e^f(x), where f(x) is any particular function, is equated to:
f'(x) . e^f(x)
where f'(x) is the differential of f(x).

y = e^ (8t sin 2t)

Let a = 8t sin 2t.

y' --> differential of y = a' e^(8t sin 2t)

a' is found by the product rule and the differential forms of sin functions.

Answer 2

Yes, thank you! I meant differentiate! (Still learning the terminology!)

In your estimation, would the final answer be: e^8t sin 2t * (8 sin 2t + 8t cos 2t)?

Have I applied the product rule correctly?

Answer 3

take the derivative of sin(2t) again

Answer 4

not "again" but you made an error