# Question Video: Differentiating Combinations of Exponential and Polynomial Functions Using the Quotient Rule Mathematics

Find the first derivative of the function π¦ = (4π)^(7π₯)/(7π₯ + 4).

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### Video Transcript

Find the first derivative of the function π¦ is equal to four times π raised to the power seven π₯ over seven π₯ plus four.

Weβre asked to find the first derivative of a function π¦, which is a function of π₯. This means weβll need to differentiate π¦ with respect to π₯. Now, a function π¦ is a rational function of the form π’ over π£, where π’ and π£ are differentiable functions of π₯. There are a couple of different methods we could use to differentiate π¦. For example, since the denominator seven π₯ plus four is a simple polynomial, we could write our function as a product.

And we could use the general power rule to differentiate seven π₯ plus four to the negative one and then use the product rule. However, since our function is a fairly simple quotient, letβs use the quotient rule. And using the notation dπ by dπ₯ is equal to π prime, for a rational function, π of π₯ which is equal to π’ of π₯ over π£ of π₯, where π’ and π£ are differentiable functions of π₯, π prime of π₯ is equal to π£ of π₯ times π’ prime of π₯ minus π’ of π₯ times π£ prime of π₯ all over π£ squared.

Now, the function π¦, if we let π’ equal four π raised to the power seven π₯ where π is Eulerβs number and π£ equal to seven π₯ plus four, to use the quotient rule, we need to find π’ prime of π₯ and π£ prime of π₯. And to differentiate four π raised to the seven π₯ with respect to π₯, we can use the known result d by dπ₯ of π raised to the power ππ₯ is equal to ππ raised to the power ππ₯, where π is a constant. In our function π’, π is equal to seven. And bringing the seven down, dπ’ by dπ₯ is four times seven times π raised to the power seven π₯, which is 28π raised to the power seven π₯.

To find π£ prime of π₯, weβre going to use the power rule for differentiation, where we note that in π£, seven π₯ is actually seven π₯ raised to the power one. Remember that the power rule says for a function of the form π times π₯ raised to the power π, where π and π are constants, d by dπ₯ is equal to π times π times π₯ raised to the power π minus one. That is, we multiply by the exponent of π₯ and subtract one from the exponent. In our case, our exponent is one, so that the derivative of seven π₯ is seven times one times π₯ raised to the power one minus one. That is seven π₯ raised to the power zero. And we know that anything to the power zero is equal to one, so thatβs equal to seven.

The derivative of our constant four is equal to zero, since a constant has no π₯ dependence, so that π£ prime of π₯ is equal to seven. So now, using our results within the quotient rule, we have seven π₯ plus four which is π£ times 28 times π raised to the power seven π₯, which is π’ prime, minus four π raised to seven π₯, which is π’, times seven, which is π£ prime, all over π£ squared. Thatβs seven π₯ plus four squared. And since four times seven is 28, we can rewrite this as 28π raised to the seven π₯ times seven π₯ plus four minus 28π raised to the power seven π₯ all over seven π₯ plus four squared.

To simplify this, we can take the common factor of 28π raised to the power seven π₯ outside some parentheses, so that we have 28π raised to the power seven π₯ times seven π₯ plus four minus one all over seven π₯ plus four squared. And since four minus one is equal to three, inside our parentheses we have seven π₯ plus three. And if we now redistribute our parentheses, we have 28π raised to the power seven π₯ times seven π₯ plus 28π raised to the power seven π₯ times three.

The first derivative of the function π¦ is equal to four times π raised to the power seven π₯ over seven π₯ plus four is therefore 196π₯ times π raised to the power seven π₯ plus 84 times π raised to the power seven π₯ all over seven π₯ plus four squared.