Find the derivatives of the functions in Exercises 19–40.

Question

y = (4x + 3)⁴(x + 1)⁻³

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says determine the derivative of Y is equal to the quantity of 5 x + 4 and quantity rates to the 5th, multiplied by the quantity of X + 2 in quantity rates to the minus 4th. So this problem wants us to find the derivative of our function Y. So we're going to be looking for Y. And that's going to be equal to the derivative. With respect to X. Of the quantity of 5 X plus 4 in quantity rates to the 5th, multiplied by the quantity of X plus 2 in quantity raised to the minus 4th power in quantity. And so, in order to take this derivative, we're going to need to recall several of our derivative rules. So we're taking a derivative of our product, we're going to want to recall our product rule. So, we call for the product rule that if we have the derivative with respect to X, Of quantity of F of X multiplied by G of X. In quantity, that that is going to be equal to F of X, multiplied by G of X. Plus F of X. Multiplied by G of X. We're also going to need to use our chain rule. So we call for the chain rule that we have the derivative with respect to X. Of F of G of X. And that is going to be equal to F of G of X. Multiplied by G of X. We're also going to need to use our constant multiple rule. So recall for the constant multiple rule that we have the derivative with respect to x of the quantity of C multiplied by X raised to the power n in quantity, and this is going to be equal to C multiplied by N, multiplied by X raised to the power of n minus 1. We're also going to need to use our power rule, so we call for the power rule that we have the derivative spec to X of X rated to the power n, and that's going to be equal to n multiplied by X rated to the quantity of n minus 1 in quantity. Then finally, we're going to need to use our constant rule, so that's going to be the derivative with respect to X of a constant C, and that is going to be equal to 0. So, we actually want to start off by looking at our product rule. And so here we want to identify our functions F of X and G of X in that product rule. So, in our case, we're gonna have F of X. B equal to the quantity of 5 X plus 4 in quantity raised to the 5th power. And G of X is going to be equal to the quantity of X + 2 in quantity raised to the minus 4th power. So, we need to calculate F of X. In order to use our product rule, so this is going to be equal to the derivative with respect to X. Of the quantity of 5 X plus 4 in quantity raised to the 5th power. And so we're going to need to apply our power rule as well as our chain rule to take this derivative. So applying our power rule, this is going to be equal to our X component, which is 5, multiplied by the quantity of 5 X plus 4 in quantity, and this is going to be raised to the quantity of 5 minus 1 in quantity. But we also need to apply our chain rule, so we need to multiply this by the derivative with respect to X of the quantity of 5 X plus 4 in quantity. So this is going to be equal to 5, multiplied by the quantity of 5 X plus 4 in quantity raised to the 4th power, and this gets multiplied by. And here we're going to take the derivative with respect to X of 5 X. I'm going to need to use our constant multiple rule. So we'll have our constant 5 multiplied by the X component, which is 1, that gets multiplied by x raised to the quantity of 1 minus 1. In quantity, then we'll have plus the derivative of 4 with respect to x. So here I have to apply our constant rule. So that's going to be 0 in quantity. So simplifying this expression, this is going to be equal to 25 multiplied by the quantity of 5 X plus 4 in quantity raised to the 4th power. We now need to find G of X, and this is going to be equal to the derivative with respect to X. Of the quantity of X + 2 in quantity raised to the minus 4th power. And so again, we're gonna be using our Power rule as well as our chain rule. And so this is going to be equal to applying our power rule first. We'll have the exponent, which is -4, multiplying the quantity of X plus 2 in quantity, raise to the quantity of -4, 1 in quantity, then applying the chain rule. This needs to be multiplied by the derivative with respect to X. Of the quantity of X + 2 in quantity. So this is going to be equal to -4, multiplied by the quantity of X plus 2 in quantity raised to the minus 5 power. And that needs to be multiplied by the derivative with respect to X. of X + 2. So we'll multiply this by the quantity. We want to take the derivative of X, I'm gonna need to use our power rule. So I'll have the exponent, which is 1 multiplied by X, raised to the quantity of 1 minus 1 in quantity. Then we'll have plus the derivative 2 with respect to X. So applying a constant rule, that's going to be equal to 0 in quantity. So simplifying this expression, this is just going to be equal to -4, multiplied by the quantity of X plus 2 in quantity raised to the minus 5 power. So now we have all the information that we need to actually calculate our derivative of Y. So Y prime? is going to be equal to and here applying our product rule we'll have F of X, which is going to be 25 multiplied by the quantity of 5 x plus 4 in quantity raised to the 4th power that gets multiplied by G of X, which is the quantity of X + 2 in quantity raised to the minus 4th power. And then we'll have plus. F of X, which is the quantity of 5 X plus 4. In quantity raised to the 5th power. And that gets multiplied by G of X, which is the quantity of -4 multiplied by the quantity of X + 2 in quantity, raised to the -5 power in quantity. So now all we need to do is simplify this expression. So I noticed that I can factor out a quantity of 5 x + 4 in quantity rates to the 4th power, and we can also factor out a quantity of X + 2 in quantity rates to the minus 5 power out of each of the terms. So this is going to be equal to. The quantity of 5 X plus 4 in quantity raised to the 4th power, multiplied by the quantity of X + 2 in quantity raised to the minus 5 power. And that gets multiplied by what's left, and in the first term, we're going to have 25 multiplied by the quantity of X + 2 in quantity. So that's going to give me 25 X. Plus 50. And then for the second term, I will have -4 multiplied by the quantity of 5 X plus 4 in quantity, and so that will give me minus 20 X. -16 in quantity. So, simplifying this expression and moving our negative exponents into the denominator, this is going to be equal to the quantity of 5 X plus 4 in quantity raised to the 4th power. Multiplied by the quantity of 5 X. Plus 34 in quantity divided by the quantity of X plus 2 in quantity raised to the fifth power. And so this here is the answer that we're looking for for this problem. So I hope this explanation has been useful, and I'll see you in the next video.

Find the derivatives of the functions in Exercises 19–40.

y = (4x + 3)⁴(x + 1)⁻³

Key Concepts

Product Rule

Chain Rule

Negative Exponents

Watch next