Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule Mathematics • Higher Education

Determine the derivative of 𝑓(𝑑) = 𝑑 sin 5πœ‹π‘‘.


Video Transcript

Determine the derivative of 𝑓 of 𝑑 equals 𝑑 sin of five πœ‹π‘‘.

We have a function in 𝑑, which is actually the product of two functions. We could say that one of the functions is 𝑑. And the other is sin of five πœ‹π‘‘. So how do we find the derivative of the product of two functions? For two functions 𝑒 and 𝑣 in π‘₯, the derivative of 𝑒 times 𝑣 is 𝑒 times the derivative of 𝑣 with respect to π‘₯ plus 𝑣 times the derivative of 𝑒 with respect to π‘₯.

Now, of course, our function is in terms of 𝑑. So we change this slightly to the derivative of 𝑒𝑣 with respect to 𝑑. And this means we can let 𝑒 be equal to 𝑑 because that’s the first function in our equation. And we can let 𝑣 be equal to sin of five πœ‹π‘‘ because that’s the second function of 𝑑.

We can see that we’re going to need to differentiate both of these with respect to 𝑑. If we differentiate 𝑒 with respect to 𝑑, we simply get one. But differentiating 𝑣 with respect to 𝑑 is a little bit trickier. We could use the chain rule. But we don’t need to. We can apply a general result. And that is if we differentiate sin of some constant of 𝑑, we get that constant multiplied by cos of that constant of 𝑑. So in this case, the derivative of sin of five πœ‹π‘‘ is five πœ‹ multiplied by cos of five πœ‹π‘‘.

All that’s left is to substitute this back into our equation for the product rule. 𝑒 multiplied by 𝑑𝑣 𝑑𝑑 is 𝑑 multiplied by five πœ‹ cos five πœ‹π‘‘. And 𝑣 multiplied by 𝑑𝑒 𝑑𝑑 is sin five πœ‹π‘‘ multiplied by one, which simplifies to five πœ‹π‘‘ multiplied by cos five πœ‹π‘‘ plus sin of five πœ‹π‘‘.

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