Question

Find the second derivative of the function. $$ h(t)=e^{t} \sin t $$

Short answer

The second derivative of the function \( h(t) = e^t \sin t \) is \( 2e^t \cos t \).

Step-by-step solution

Calculate the First Derivative

The first derivative of \( h(t) \) can be calculated using the product rule, which states that the derivative of two functions multiplied together is the first function times the derivative of the second plus the second function times the derivative of the first. Hence, differentiating \( h(t) \), \( h'(t) \) = \( e^t \sin t \) + \( e^t \cos t \) = \( e^t (\sin t + \cos t) \).

Calculate the Second Derivative

The second derivative of \( h(t) \) is the derivative of the first derivative. Again using the product rule, \( h''(t) \) can be found: \( h''(t) \) = \( e^t (\cos t - \sin t) + e^t (\sin t + \cos t) \) = \( e^t (2\cos t) \) = \( 2e^t \cos t \).

Simplify the Expression

No simplification is needed in this particular case as the second derivative is as simplified as it can get in this context, thus obtaining: \( h''(t) = 2e^t \cos t \).