# Definition of the derivative to find the derivative of x^(1/3)

• Martinc31415
In summary, the student is struggling with finding the derivative of x^(1/3) using the definition of the derivative. They have attempted to use the difference of cubes formula, but are unsure of how to proceed. They are then given guidance on how to use the conjugate of the difference of cube roots to simplify the problem.
Martinc31415

## Homework Statement

Use the definition of the derivative to find the derivative of x^(1/3)

## The Attempt at a Solution

[(x+h)^(1/3) - x^(1/3)]/h

I do not know where to go from here. If it were a square root I could conjugate.

Martinc31415 said:

## Homework Statement

Use the definition of the derivative to find the derivative of x^(1/3)

## The Attempt at a Solution

[(x+h)^(1/3) - x^(1/3)]/h

I do not know where to go from here. If it were a square root I could conjugate.
Hello Martinc31415. Welcome ton PF !

The difference of cubes can be factored, $a^3-b^3=(a-b)(a^2+ab+b^2)\,.$

So, suppose you have the difference of cube roots, $\displaystyle P^{1/3}-Q^{1/3}$. In this case, $\displaystyle P^{1/3} = a\,\ \text{ and }\ Q^{1/3} = b\,.$

Multiplying $\displaystyle \left(P^{1/3}-Q^{1/3}\right)$ by $\displaystyle \left(P^{2/3}+P^{1/3}Q^{1/3}+Q^{2/3}\right)$ will give $\displaystyle \left(P^{1/3}\right)^3-\left(Q^{1/3}\right)^3=P-Q\,.$

Thus, $\displaystyle \left(P^{2/3}+P^{1/3}Q^{1/3}+Q^{2/3}\right)$ acts as the "conjugate" for $\displaystyle \left(P^{1/3}-Q^{1/3}\right)\,.$

oohh...

I would not have thought of that, ever!

Thanks a ton.

## 1. What is the definition of the derivative?

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

## 2. How is the derivative calculated?

The derivative is calculated using the limit definition: f'(x) = lim h->0 (f(x+h)-f(x))/h. This involves finding the tangent line to the function at a specific point and determining its slope.

## 3. How is the derivative of x^(1/3) found?

The derivative of x^(1/3) can be found using the power rule for derivatives, which states that the derivative of x^n is nx^(n-1). In this case, n = 1/3, so the derivative is (1/3)x^(1/3-1) = 1/(3x^(2/3)).

## 4. What is the importance of finding the derivative of a function?

The derivative of a function is important because it allows us to analyze the rate of change of the function at a specific point. This can help us understand the behavior of the function and make predictions about its future values.

## 5. How is the derivative used in real life applications?

The derivative is used in a variety of real life applications, including physics, economics, and engineering. It is used to calculate velocities, acceleration, and rates of change in various systems. In economics, it can be used to determine the marginal cost and revenue of a product. In engineering, it is used to optimize designs and determine the efficiency of systems.

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