Question

Use the preceding two derivative formulas to make a conjecture about a formula for $\frac{d}{dx}\left(x^n\right)$, whererole="math" $n$ is a positive integer.

Short answer

From the previous results, we have the following required conjecture :-

$\frac{d}{dx}\left(x^n\right)=n*x^{n-1}$, where $n$ is a positive integer.

Step-by-step solution

Step 1. Given Information

We have given the following function :-

$x^n$.

In previous two exercises, we find that :-

$\frac{d}{dx}\left(x^4\right)=4x^3$and $\frac{d}{dx}\left(x^8\right)=8x^7$.

By using both of these results, we have to find a conjecture for the derivative of the given function.

Step 2. Make Conjecture

In previous exercises we proved that :-

$\frac{d}{dx}\left(x^4\right)=4x^3$and $\frac{d}{dx}\left(x^8\right)=8x^7$

Both of these functions, $x^4$and role="math" $x^8$are of type role="math" localid="1648387831171" $x^n$, where $n$is a positive integer.

In both of these we can see that while we taking derivative, then in the result the power of the function become multiplier of the function and power reduced by one.

From this discussion, we have the following conjecture, about the derivative of the function $x^n$:-

$\frac{d}{dx}\left(x^n\right)=n*x^{n-1}$, where $n$ is a positive integer.