Derivative of function with radicals

In summary, the conversation is about finding the derivative of f(x) = x + √x and the steps involved in solving it. The person got stuck at a certain step and needed help understanding how to proceed. The expert summarizer provided a step-by-step summary of the solution, including the use of rationalization and taking the limit, and clarified any confusion the person had.
  • #1
PhysChem
3
0
I've been having trouble figuring out how to find the derivative of f(x) = x + √x

The farthest I got was:

[(x+h) + √(x+h) - (x+√x)]/h =

[h + √(x+h) - √x] / h

I got stuck here because I'm not sure how to cancel out h in numerator and denominator (if i can even do that at this stage) or multiply the whole equation by the conjugate. For some reason I'm stumped on what seems like simple algebra.
 
Physics news on Phys.org
  • #2
PhysChem said:
I've been having trouble figuring out how to find the derivative of f(x) = x + √x

The farthest I got was:

[(x+h) + √(x+h) - (x+√x)]/h =

[h + √(x+h) - √x] / h

I got stuck here because I'm not sure how to cancel out h in numerator and denominator (if i can even do that at this stage) or multiply the whole equation by the conjugate. For some reason I'm stumped on what seems like simple algebra.
That is, of course, [itex]1+ \frac{\sqrt{x+h}- \sqrt{x}}{h}[/itex]. The "1" is of course, the derivative of "x". All that is left is the last fraction. To do that "rationalize" the numerator: multiply both numerator and denominator by [itex]\sqrt{x+h}+ \sqrt{x}[/itex] to get
[tex]\frac{\sqrt{x+h}- \sqrt{x}}{h}\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}[/tex]
[tex]= \frac{(x+ h)- x}{h(\sqrt{x+h}+\sqrt{x})}= \frac{1}{\sqrt{x+h}+ \sqrt{x}}[/tex]

Now, it's easy to take the limit as h goes to 0.
 
  • #3
I see, thank you for the help! the "1 + (fraction)" was what I missed.

Does this mean my second step [h + √(x+h) - √x] / h was wrong?

I'm still not sure how to arrive at your first step, the "1 + (fraction)". However, everything else makes sense now!
 
  • #4
PhysChem said:
I see, thank you for the help! the "1 + (fraction)" was what I missed.

Does this mean my second step [h + √(x+h) - √x] / h was wrong?

I'm still not sure how to arrive at your first step, the "1 + (fraction)". However, everything else makes sense now!

No your second step isn't wrong. HallsofIvy's first step comes from your second step:
[tex]\frac{h + \sqrt{x+h}- \sqrt{x}}{h}[/tex]
[tex]= \frac{h}{h} + \frac{\sqrt{x+h}- \sqrt{x}}{h}[/tex]
[tex]= 1 + \frac{\sqrt{x+h}- \sqrt{x}}{h}[/tex]
 

FAQ: Derivative of function with radicals

1. What is the derivative of a function with a radical?

The derivative of a function with a radical is the slope of the tangent line to the function at a specific point. It represents the rate of change of the function at that point.

2. How do you find the derivative of a function with a radical?

To find the derivative of a function with a radical, you can use the power rule or the chain rule. First, rewrite the function with the radical as a fractional exponent, then apply the appropriate rule to find the derivative.

3. Can you give an example of finding the derivative of a function with a radical?

Yes, for example, if we have the function f(x) = √x, we can rewrite it as f(x) = x1/2. Then, using the power rule, we can find the derivative as f'(x) = (1/2)x-1/2 = 1/(2√x).

4. Why is finding the derivative of a function with a radical important?

Finding the derivative of a function with a radical is important because it allows us to analyze the behavior of the function and make predictions about its rate of change. It is also a fundamental concept in calculus and is used in many real-world applications, such as physics and engineering.

5. Are there any special cases when finding the derivative of a function with a radical?

Yes, there are a few special cases to consider when finding the derivative of a function with a radical, such as when the radical is raised to a power other than 1/2 or when the function contains multiple radicals. In these cases, different rules may need to be applied to find the derivative.

Similar threads

B Why does the Limit Process give an Exact Slope?
2
Replies
53
Views
2K
I From a proof on directional derivatives
Replies
9
Views
2K
I Exploring the Derivative of y(x,t) in Quantum Mechanics
Replies
1
Views
1K
A Multi-variable function depending on the Heaviside function
Replies
1
Views
2K
I Derivatives of Standard Functions
Replies
17
Views
2K
MHB Trouble finding the derivative of a fraction using four step process
Replies
2
Views
1K
B How do I invert this exponential function?
Replies
3
Views
3K
I Deriving a function from within an integral with a known solution
Replies
6
Views
2K
I Limits to directly check second order differentiability
Replies
1
Views
1K
A How to take the spatial derivative of quaternions
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top