Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule | Nagwa Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule | Nagwa

# Question Video: Differentiating Functions Involving Trigonometric Ratios Using the Product Rule Mathematics • Second Year of Secondary School

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Determine the derivative of π(π‘) = π‘ sin 5ππ‘.

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### Video Transcript

Determine the derivative of π of π‘ equals π‘ sin of five ππ‘.

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

Now, of course, our function is in terms of π‘. So we change this slightly to the derivative of π’π£ with respect to π‘. And this means we can let π’ be equal to π‘ because thatβs the first function in our equation. And we can let π£ be equal to sin of five ππ‘ because thatβs the second function of π‘.

We can see that weβre going to need to differentiate both of these with respect to π‘. If we differentiate π’ with respect to π‘, we simply get one. But differentiating π£ with respect to π‘ is a little bit trickier. We could use the chain rule. But we donβt need to. We can apply a general result. And that is if we differentiate sin of some constant of π‘, we get that constant multiplied by cos of that constant of π‘. So in this case, the derivative of sin of five ππ‘ is five π multiplied by cos of five ππ‘.

All thatβs left is to substitute this back into our equation for the product rule. π’ multiplied by ππ£ ππ‘ is π‘ multiplied by five π cos five ππ‘. And π£ multiplied by ππ’ ππ‘ is sin five ππ‘ multiplied by one, which simplifies to five ππ‘ multiplied by cos five ππ‘ plus sin of five ππ‘.

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