# Question Video: Finding the Derivative of a Function Involving Trigonometric and Exponential Functions Using the Product Rule Mathematics • Higher Education

Differentiate π(π₯) = π^(π₯) cos π₯.

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### Video Transcript

Differentiate π of π₯ is equal to π to the power of π₯ multiplied by the cos of π₯.

Weβre given a function π of π₯, and we need to differentiate this function. And since π is a function of π₯, this means we need to differentiate this with respect to π₯. To do this, letβs start by looking at our function π of π₯. We can see itβs the product of two functions. Itβs π to the power of π₯ multiplied by the cos of π₯. And we know we can find the derivative of the products of two functions by using the product rule. Letβs start by recalling the product rule.

The product rule tells us if we have a function π of π₯ which is the product of two differentiable functions π’ of π₯ times π£ of π₯, then π prime of π₯ is equal to π’ prime of π₯ times π£ of π₯ plus π£ prime of π₯ times π’ of π₯. And we can see this is whatβs happening in this question. We could set π’ of π₯ to be π to the power of π₯ and π£ of π₯ to be the cos of π₯. And we know both of these are differentiable. So we can find π prime of π₯ by using the product rule.

Now, to use the product rule, we see we need to find expressions for π’ prime of π₯ and π£ prime of π₯. Letβs start with π’ prime of π₯. Since π’ of π₯ is π to the power of π₯, π’ prime of π₯ will be the derivative of π to the power of π₯ with respect to π₯.

To evaluate this derivative, we need to recall the derivative of the exponential function π to the power of π₯ with respect to π₯ is equal to itself, π to the power of π₯. So by applying this, we just get π’ prime of π₯ is equal to π to the power of π₯.

We now need to find an expression for π£ prime of π₯. Since π£ of π₯ is the cos of π₯, this will be the derivative of the cos of π₯ with respect to π₯. To evaluate this, we need to recall one of our standard trigonometric derivative results. The derivative of the cos of π₯ with respect to π₯ is equal to negative the sin of π₯. So by applying this, we get that π£ prime of π₯ is equal to negative the sin of π₯. Weβre now ready to use the product rule to help us find an expression for π prime of π₯. Remember, itβs equal to π’ prime of π₯ times π£ of π₯ plus π£ prime of π₯ times π’ of π₯.

Substituting in our expressions for π’ of π₯, π£ of π₯, π’ prime of π₯, and π£ prime of π₯, we get that π prime of π₯ is equal to π to the power of π₯ times the cos of π₯ plus negative the sin of π₯ multiplied by π to the power of π₯. And we could leave our answer like this. However, we can also simplify by taking out the common factor of π to the power of π₯.

And this gives us our final answer: π to the power of π₯ times the cos of π₯ minus the sin of π₯. Therefore, given π of π₯ is equal to π to the power of π₯ multiplied by the cos of π₯, we were able to differentiate this with respect to π₯ by using the product rule. We got π prime of π₯ is equal to π to the power of π₯ times the cos of π₯ minus the sin of π₯.