Derivative of (2x^2 + 1)x^(1/2) using Product Rule | Homework Question

In summary, to find the derivative of y= (2x2 + 1) x1/2, you can use the Product Rule. After differentiating, you may get stuck at \frac{4x3/2+2x2+1}{2\sqrt{x}}. However, the correct solution is \frac{10x2+1}{2\sqrt{x}}. The problem may be in algebra, as the left side should be (x^1/2)4x instead of 4x^3/2. By multiplying
  • #1
dh743
15
0

Homework Statement


Find the derivative of y= (2x2 + 1) x1/2


Homework Equations


Product Rule

The Attempt at a Solution


After differentiating, I eventually get stuck at:
[tex]\frac{4x3/2+2x2+1}{2\sqrt{x}}[/tex]

The given solution is [tex]\frac{10x2+1}{2\sqrt{x}}[/tex]
 
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  • #2
the problem is in your algebra, on the left you have, after differentiating, (x^1/2)4x, now when putting it over a common denom, you get 2x^1/2[(x^1/2)4x] which becomes 2x times 4x... from here i think u can finish it correctly
edit: or the way you've done it, (4x^3/2) which is the left, you need to times it by the common denom, so 4x^3/2 times 2x^1/2
 
Last edited:

FAQ: Derivative of (2x^2 + 1)x^(1/2) using Product Rule | Homework Question

What is the Product Rule for finding derivatives?

The Product Rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

How do you apply the Product Rule to find the derivative of a specific function?

To apply the Product Rule, you first identify the two functions that are being multiplied together. Then, you take the derivative of each function separately, and plug them into the formula for the Product Rule.

What are the steps for finding the derivative of (2x^2 + 1)x^(1/2) using the Product Rule?

The steps for finding the derivative using the Product Rule are as follows:

  1. Identify the two functions being multiplied together (in this case, 2x^2 + 1 and x^(1/2)).
  2. Take the derivative of each function separately.
  3. Plug the derivatives into the formula for the Product Rule: (f'g + fg').
  4. Simplify the resulting equation.

What is the derivative of (2x^2 + 1)x^(1/2)?

The derivative of (2x^2 + 1)x^(1/2) is equal to (4x + 1)/2x^(1/2).

Why is it important to know how to find derivatives using the Product Rule?

Knowing how to find derivatives using the Product Rule is important because it allows us to find the rate of change of a function that is a product of two other functions. This is a common occurrence in real-world applications, such as in physics and economics, and being able to find derivatives using the Product Rule helps us to better understand and analyze these situations.

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