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

Question

y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

Accepted Answer

Hello. In this video, we are going to be finding the derivative of the given function. So we are given Y is equal to the quantity 7 minus 3 X raised to the -2 plus 1/5 multiplied by the quantity, 3 divided by X + 4 raised to the third power. Now, in order to take the derivative of this function, we are going to need to use the chain rule. The chain rule allows us to take the derivative of a composite function, which is a function inside of a function. Now, the derivative or the chain rule states that we take the derivative of the outermost function, while keeping the innermost function the same, then we multiply this by the derivative of the innermost function. So in order to take the derivative of this Y function, we are going to need to take the derivative of each term independently. Let's go ahead and start with 7 minus 3X, raise to the negative second power. Now, since this quantity is raised to the negative second power, our outermost function is going to need to use the power rule. So applying power rule to the outermost function, which is -2, is going to give us -2 multiplied by 7 minus 3X since we have to keep the inside the same, and then we reduce the power by 1, which is going to give us -3. Now, by the chain rule, we have to then multiply this by the derivative of the innermost function. Now, the derivative of 7 minus 3X is going to be -3. So this is going to be the derivative of the first part of Y. Now, if we move to the second part of Y, we have 15th, which is a coefficient multiplied by 3 divided by X + 4 raised to the 3 power. Now, in this case here, our outermost function is going to be this exponent of 3, which is going to require power rule. So that is going to be 15th multiplied by 3, then multiplied by 3 divided by X + 4, and then we have to reduce this power of 3 by 1. That is going to leave us with a power of 2. Then again, by the chain rule, we have to multiply the derivative of our innermost function, which is going to be the derivative of 3 divided by X plus 4. By the quotient rule that is going to give us -3 divided by X squared. So, if we take these two pieces and we put them together, Y is equal to -2. Multiplied by -3, multiplied by 7, minus 3 X rated to the negative 3 power, then we are going to add 1/5 multiplied by 3, multiplied by 3 divided by X + 4 quantity squared, then multiplied by -3 divided by X2. Now what we need to do is we just need to simplify the function. -2 multiplied by -3, is going to give us 6. And if we want to turn this exponent positive, we are going to place 7 minus 3 X under 1. And multiplying that together is going to give us 6 divided by 7 minus 3X raise to the third power. Now, for the second part of this function, 1/5th multiplied by 3 is going to give us 350s. But then if we multiply this by -3 divided by X2, that is going to give us negative. 9 divided by 5 X squared, and this will be multiplied by 3 divided by X plus 4 quantity squared. Now, this is going to be the simplest form of the derivative, so this is going to be our derivative by using the chain rule. So I hope this video helps you in understanding how to use the chain rule, and I will go ahead and see you all in the next video.

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

y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴

Key Concepts

Derivative

Chain Rule

Power Rule

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