Video: Differentiating Trigonometric Functions Using the Product Rule

If π¦ = 8π₯ cos 6π₯, find dπ¦/dπ₯.

02:43

Video Transcript

If π¦ is equal to eight π₯ times the cos of six π₯, find the derivative of π¦ with respect to π₯.

We need to find an expression for the derivative of π¦ with respect to π₯, and we can see that weβre given π¦ as the product of two functions. Itβs the product of eight π₯ and the cos of six π₯, so we can try and find dπ¦ by dπ₯ by using the productβs rule. We recall the product rule tells us if π¦ is the product of two differentiable functions, π’ of π₯ times π£ of π₯, then the derivative of π¦ with respect to π₯ is equal to π’ of π₯ times the derivative of π£ with respect to π₯ plus π£ of π₯ times the derivative of π’ with respect to π₯.

In our case, π¦ is equal to eight π₯ multiplied by the cos of six π₯, so weβll set π’ of π₯ to be eight π₯ and π£ of π₯ to be the cos of six π₯. And we know how to differentiate both of these expressions, so we can do this by using the product rule. To use the product rule, we need expressions for dπ’ by dπ₯ and dπ£ by dπ₯, so letβs calculate these first. Letβs start with dπ’ by dπ₯. Thatβs the derivative of eight π₯ with respect to π₯. We could do this by using the power of the differentiation. We would rewrite eight π₯ as eight π₯ to the first power and then multiply by our exponent of π₯ and then reduce this exponent by one.

However, we can also do this by noticing eight π₯ is a linear function. And we know the slope of a linear function is just the coefficient of π₯, which in this case is eight. So we have dπ’ by dπ₯ is equal to eight. We also need to find an expression for dπ£ by dπ₯. Thatβs the derivative of the cos of six π₯ with respect to π₯. And to differentiate this expression, we need to recall one of our standard rules for differentiating trigonometric functions. We know for any real constant π, the derivative of the cos of ππ₯ with respect to π₯ is equal to negative π times the sin of ππ₯. In this case, our value of π is equal to six, so we get the derivative of the cos of six π₯ with respect to π₯ is equal to negative six times the sin of six π₯.

Now that weβve found expressions for dπ’ by dπ₯ and dπ£ by dπ₯, we can use the product rule to help us find dπ¦ by dπ₯. Itβs equal to π’ of π₯ times dπ£ by dπ₯ plus π£ of π₯ times dπ’ by dπ₯. Substituting in our expressions for π’ of π₯, π£ of π₯, dπ£ by dπ₯, and dπ’ by dπ₯, we get that dπ¦ by dπ₯ is equal to eight π₯ times negative six sin of six π₯ plus the cos of six π₯ all multiplied by eight. And if we evaluate and rearrange this equation, we get negative 48π₯ times the sin of six π₯ plus eight times the cos of six π₯. And this is our final answer.

Therefore, we were able to show if π¦ is equal to eight π₯ times the cos of six π₯, then the derivative of π¦ with respect to π₯ is equal to negative 48π₯ times the sin of six π₯ plus eight multiplied by the cos of six π₯.

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