Find the derivatives of the functions in Exercises 1–42.

Question

_____

𝔂 = / x² + x

√ x²

Accepted Answer

Hello, in this video, we are going to be calculating the derivative of the given function. The function given to us is Y is equal to the quantity X cubed minus X2, divided by X cubed, raised the power of 14th. Now, before we take the derivative, let's go ahead and simplify the function. If we look at the innermost portion of the function, notice that we can separate X cubed minus X2, divided by X cubed into a difference of two functions. We can rewrite the innermost part of Y to be X cubed, divided by X cubed minus X2 divided by X cubed. This will further simplify to be 1 minus 1 divided by X. So we can rewrite the original function to be Y is equal to 1 minus 1 divided by X raised to the power of 1/4. Now what we can go ahead and do is take the derivative. Notice that our function Y is in the form of a composite function. The innermost function is 1 minus 1 divided by X, and the outermost function is going to be the exponent of 14th. So in order to take the derivative, we will need to use the chain rule. So, in order to use the chain rule, what we will do is we will take the derivative of the outermost function, while keeping the innermost function the same first. So, we will go ahead and use the power rule on the exponent of 14th. By using the power rule, the outermost derivative is going to be 14th, multiplied by the quantity, 1 minus 1 divided by X, raise to the power of -3 divided by 4. Then by the chain rule, we will go ahead and multiply the derivative of the outermost function by the derivative of the innermost function. So, the derivative of the innermost function will be the derivative of 1 which is 0, minus the derivative of 1 divided by X, which will be negative 1 divided by X2. So the derivative of the innermost function is going to be negative. 1 divided by X2. Now what we need to do is we need to simplify the derivative. -1 multiplied by -1 divided by X2, will be a positive quantity. So we have 14th. Multiplied by 1 minus 1 divided by X, raises the power of -3/4, multiplied by positive, 1 divided by X2. Next, we want to go ahead and turn this negative exponent into a positive exponent. In order to do this, we can take the entirety of this quantity, 1 minus 1 divided by X, raise to the power of -3/4, and move it to the denominator of 1/4. That will allow us to rewrite the derivative to be 1 divided by 4 multiplied by the quantity, 1 minus 1 divided by X, raises the power of 3/4, and this is still multiplied by 1 divided by X2. Next, what we can do is we can multiply the fractions together, and that is going to give us a final derivative of 1 divided by 4 X squad, multiplied by the quantity, 1 minus 1 divided by X raised to the power of 3/4, and this is going to be the simplest form of the of the derivative by using the chain rule. And with that being said, the solution to this problem is going to be a. So I hope this video helps you in understanding on 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 1–42.
_____
𝔂 = / x² + x
√ x²

Key Concepts

Derivatives

Quotient Rule

Simplifying Functions

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