# Mathematical Induction: Power Rule for Differentiation

• BrianMath
In summary, the proof uses the Product Rule for differentiation and mathematical induction to show that \frac{d}{dz}z^n = nz^{n-1}\;\;\; \forall n\in\mathbb{N}. It begins by proving the base case for n=1 and then assumes the statement is true for n=k. From there, it shows that it is also true for n=k+1, completing the proof by induction.
BrianMath

## Homework Statement

Prove that
$$\frac{d}{dz}z^n = nz^{n-1}\;\;\; \forall n\in\mathbb{N}$$
using the Product Rule for differentiation and mathematical induction.

## Homework Equations

$$\frac{d}{dz} f(z) = \lim_{\Delta z\to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z}$$
$$\frac{d}{dz}[f(z)g(z)] = f\,'(z)g(z) + g'(z)f(z)$$

## The Attempt at a Solution

Let n = 1:
$$\frac{d}{dz} z = \lim_{\Delta z \to 0} \frac{z + \Delta z - z}{\Delta z} = \lim_{\Delta z \to 0} \frac{\Delta z}{\Delta z} = \lim_{\Delta z \to 0} 1 = 1$$

Assume true for n = k:
$$\frac{d}{dz} z^k = kz^{k-1}$$

Let n = k + 1:
$$\frac{d}{dz} z^{k+1} = \frac{d}{dz}[z^k\cdot z] = \frac{d}{dz} (z^k) \cdot z + \frac{d}{dz} (z) \cdot z^k = kz^{k-1}\cdot z + z^k = kz^k + z^k = (k+1)z^k = (k+1)z^{(k+1)-1}$$

Like my previous topic, I'm pretty sure I have the proof correct, but I need to make sure that it is written out correctly. I don't want to be teaching myself the wrong way to lay out proofs.

Looks fine to me.

## 1. What is the Power Rule for Differentiation?

The Power Rule for Differentiation is a formula used in calculus to find the derivative of a polynomial function. It states that the derivative of a term with a variable raised to a power is equal to the power multiplied by the coefficient of the term, and the power of the variable decreased by one.

## 2. How do you use the Power Rule for Differentiation?

To use the Power Rule for Differentiation, you first identify the term in the function with a variable raised to a power. Then, you multiply the power by the coefficient of the term and decrease the power of the variable by one. This process is repeated for each term in the function.

## 3. Can the Power Rule for Differentiation be applied to all polynomial functions?

Yes, the Power Rule for Differentiation can be applied to all polynomial functions, as long as the function is continuously differentiable. This means that the function has a well-defined derivative at every point in its domain.

## 4. What is the difference between the Power Rule for Differentiation and the Power Rule for Integration?

The Power Rule for Differentiation is used to find the derivative of a polynomial function, while the Power Rule for Integration is used to find the antiderivative or integral of a polynomial function. The Power Rule for Integration is essentially the reverse of the Power Rule for Differentiation.

## 5. Can the Power Rule for Differentiation be used for functions with non-integer exponents?

Yes, the Power Rule for Differentiation can also be applied to functions with non-integer exponents, as long as the exponent is a constant. In this case, the power is multiplied by the coefficient of the term, and the variable is raised to the power minus one, just like with integer exponents.

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