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Find the derivative of the function.

$ f(t) = e^{at} \sin bt $

$e^{a t}(b \cos b t+a \sin b t)$

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Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

let's find the derivative of this function and this function is actually a product e to the A. T is one factor and sign of BT is the other factor. So we're going to use the product rule so the derivative will be f prime of tea equals the first each of the 80 times the derivative of the second. Now to use to find the derivative of the second, we're going to use the chain rule because we have an inside function BT and we haven't outside function sign. So the derivative of sign is co sign so we would have co sign of BT times the derivative of the inside the derivative A bt would be be So here's what we have so far the first times the derivative of the second. Now we need plus the second sign of BT times the derivative of the first, so the derivative of each of the 80 that would also use the chain rule. So the outside part would be each of the 80 times the derivative of the inside the derivative of 80 would be a Okay, so that was the second times the derivative. The second time's a derivative of the first. Now, as we simplify this answer, let's look at the first term and the second term and see if they have any common factors. They both have e to the a t. So let's factor that out each of the 80 times What? What do we have left in the first term? We have the be co sign Bt and what do we have left in the second term? We have the sign of BT and the A so there's are derivative.

Oregon State University