# 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₯) csc π₯.

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### Video Transcript

Differentiate π of π₯ is equal to π to the power of π₯ times the csc of π₯.

We need to differentiate the function π of π₯. We can see that π is a function of π₯. So, we need to differentiate this with respect to π₯. To do this, we need to notice that π of π₯ is the product of two functions. Itβs π to the power of π₯ multiplied by the csc of π₯. And in fact, itβs the product of two functions which we know how to differentiate. This means we can do this by using the products rule. So, letβs recall what the product rule tells us.

The product rule tells us if π of π₯ is equal to 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₯. To use the product rule in this case, weβre going to need to set π’ of π₯ equal to π to the power of π₯ and π£ of π₯ equal to the csc of π₯. This then means that π of π₯ is equal to π’ of π₯ multiplied by π£ of π₯.

This means to use the product rule, weβre now going to need to find expressions for π’ prime of π₯ and π£ prime of π₯. Letβs start with π’ prime of π₯. Thatβs the derivative of π to the power of π₯ with respect to π₯. And this is a standard derivative result we should commit to memory. The derivative of the exponential function π to the power of π₯ with respect to π₯ is just equal to itself, π to the power of π₯. So, π’ prime of π₯ is just π to the power of π₯.

We now need to find an expression for π£ prime of π₯. Thatβs the derivative of the csc of π₯ with respect to π₯. Thereβs a few different ways we could evaluate this derivative. For example, we could write the csc of π₯ as one divided by the sin of π₯ and then use either the quotient rule or the general power rule. However, itβs easier to just recall that the derivative of the csc of π₯ with respect to π₯ is equal to negative the csc of π₯ times the cot of π₯. So, using this, we now have π£ prime of π₯ is equal to negative the csc of π₯ times the cot of π₯.

Now that we found expressions for π’ prime of π₯ and π£ prime of π₯, weβre ready to find an expression for π prime of π₯ by using the product rule. 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 csc of π₯ plus negative one times the csc of π₯ multiplied by the cot of π₯ times π to the power of π₯.

And thereβs a few ways we can simplify this. Weβll take out the common factor of π to the power of π₯ and the common factor of the csc of π₯. And this leaves us with our final answer, π to the power of π₯ times the csc of π₯ multiplied by one minus the cot of π₯.

Therefore, given π of π₯ is equal to π to the power of π₯ times the csc of π₯, we were able to use the product rule to show that π prime of π₯ is equal to π to the power of π₯ times the csc of π₯ multiplied by one minus the cot of π₯.