# Help finding the derivative of rational/radical function

• jmanna98
In summary, the conversation discusses finding the derivative of F(x) = (-1/sqrt(2x)) +2x and the steps involved in simplifying the difference quotient. The speaker is having trouble with rationalizing the numerator and finding the correct LCD. They are advised to break the function into two pieces to make the problem more manageable.
jmanna98
Please help me break down the first couple parts of this derivative question. This question gets a bit ugly:

Find the derivative of F(x)=(-1/sqrt(2x)) +2x

You can rewrite F(x) = 2x - (2x)^(-1/2)

Try taking the derivative of this expression.

ya but then how do you multiply out (x+deltax)^.5?

So, do you need to find the derivative by finding the limit: $\displaystyle \lim_{\Delta x \to 0} \frac{F(x+\Delta x)-F(x)}{\Delta x}\,?$

If so, you'll find it handy to rational the numerator.

Yes as delta x approaches zero. I know there is goig to be some conjugate or LCD stuff going on but i got stuck

So far you haven't shown any work at all. What have you tried to do and where, exactly, do you have a problem?

The first thing I did was sub the function into the derivative formula which made a huge mess of a problem to simplify.

[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax.

I am a little rusty on working with radicals and tried a few things that ended up in a mess but I am thinking that I should simplify the numerator of this first by finding the LCD of the rational expressions in the numberator of the whole problem. LCD:(sqrt2(x+deltax))(sqrt2x)? Then multiply by the conjugate? I sort of feel on the right track but at the same time I feel that my LCD is incorrect for some reason.

jmanna98 said:
The first thing I did was sub the function into the derivative formula which made a huge mess of a problem to simplify.

[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax.

I am a little rusty on working with radicals and tried a few things that ended up in a mess but I am thinking that I should simplify the numerator of this first by finding the LCD of the rational expressions in the numberator of the whole problem. LCD:(sqrt2(x+deltax))(sqrt2x)? Then multiply by the conjugate? I sort of feel on the right track but at the same time I feel that my LCD is incorrect for some reason.
Let's look at your difference quotient in LaTeX.
[(-1/sqrt2(x+deltax)) +2(x+deltax)] - [(-1/sqrt2x)+2x] all over deltax
$$\frac{\displaystyle \frac{-1}{\sqrt{2(x+\Delta x)}}-\left(\frac{-1}{\sqrt{2(x)}}+2x\right)}{\Delta x}\quad\to\quad \frac{\displaystyle \frac{-1}{\sqrt{2(x+\Delta x)}}-\frac{-1}{\sqrt{2(x)}}}{\Delta x}+\frac{2(x+\Delta x) -2x}{\Delta x}$$

Instead of trying to find the limit of all of the parts of F(x) at one time, break F(x) into two pieces: 2x and the radical. Since the derivative of a sum is the sum of the derivatives of the components, you can calculate the two limits and add them together.

## What is the definition of a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.

## What is a rational function?

A rational function is a function that can be expressed as the ratio of two polynomials. It can also be written in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomials.

## What is a radical function?

A radical function is a function that contains a radical expression, such as a square root, cube root, or higher root. It can also be written in the form of f(x) = √(x) or f(x) = ∛(x).

## How do you find the derivative of a rational function?

To find the derivative of a rational function, you can use the quotient rule, which states that the derivative of f(x)/g(x) is equal to (f'(x)g(x) - f(x)g'(x))/[g(x)]^2. Alternatively, you can rewrite the function as f(x) = p(x)q(x)^-1 and use the power rule to find the derivative.

## How do you find the derivative of a radical function?

To find the derivative of a radical function, you can use the chain rule, which states that the derivative of √(f(x)) is equal to 1/2√(f(x)) * f'(x). Similarly, for ∛(f(x)), the derivative would be 1/3∛(f(x)) * f'(x). You can also rewrite the function using rational exponents and use the power rule to find the derivative.

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