Evaluate and simplify y'.

y = 4x⁴ ln x −...

Question

Evaluate and simplify y'.

y = 4x⁴ ln x − x⁴

Accepted Answer

Welcome back, everyone. Determine the derivative of Y with respect to X, given that Y equals 2 X 2 LN of X2 minus 3 X2 and were given for answer choices A Y equals 2. X multiplied by 4 L X minus 1 B 2 X multiplied by 4 LNX plus 1 C 4 X multiplied by 2 LNX minus 1 and D 4 X multiplied by 2 LNX plus 1. So let's analyze the given function. It says that Y equals 2 X 2 L and X2. Minus 3 x 2d and here to simplify while we can apply the properties of logarithms we can rewrite this as 2 x 2 multiplied by 2 l and X right because we can bring down the exponent of X in front of the log and then we are subtracting 3 x 2. So this gives us y equals 4 x 2 l and X minus 3 x 2. Moreover, we can factor it out and we can say that x2d can be taken out, which gives us X2d imperies. 4 L and X minus 3. So now from here we can start differentiating. We have Y equals the derivative of x2 multiplied by 4 L and X minus 3. And as we can see we have a product of two functions. Let's call one of them F. F is going to be the first function which is X2. And the second one in the product is G, which is 4L and X minus 3. Let's recall the product rule. If we want to differentiate F multiplied by G, this gives us F G plus FG. So we want to find two derivatives, right? The first one is F. So we want to differentiate X2. According to the power rule, we bring down the exponent, that's 2, and the original exponent of X decreases by 1 unit, so 2 minus 1 gives us 1. So the derivative of X2d is 2 X. What about G? Well, the derivative of LN of X based on the table of basic derivatives is 1 divided by X, and 1 divided by X multiplied by 4 gives us 4 divided by X, and the derivative of -30 because it is a constant. So now let's apply the product rule. We first will take F which is 2 X. We multiply that by G, which is 4 L and X minus 3. And then we add F multiplied by G. So F is X2 and G is 4 divided by X. And from here we can simplify, so we get 2 X. And we can distribute, right, so 2 X multiplied by 4 LN of X is simply 8. XL and X minus We multiplied by 2 gives us 6, and then we have X, and then we are adding X squared multiplied by 4 divided by X, which is 4 X. So overall, this is 8 XL and X minus 2 X. And now let's factor out 2 X. So we get 2 X in 4 LNX minus 1. That's our final answer, and if we analyze the answer choices. We can conclude that the correct answer corresponds to the choice A. Thank you for watching.

Evaluate and simplify y'.

y = 4x⁴ ln x − x⁴

Key Concepts

Differentiation

Product Rule

Logarithmic Differentiation

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