# Derivatives (Part III) - Calculus Tutorials

**Notation:** The derivative $f'(x)$ of a function $f(x)$ is also denoted as

This notation comes from the fact that when you compute the derivative, you compute

\[\frac{df}{dx}(x_0) = \displaystyle\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}\]The term $f(x)-f(x_0)$ is usually referred as $\Delta f$, and the term $x-x_0$ is referred as $\Delta x$. So, sometimes, in some books (specially Physics books) you're going to find the definition

\[\frac{df}{dx}(x_0) = \displaystyle\lim_{\Delta x\to 0} \frac{\Delta f}{\Delta x}\]**Theorems to calculate Derivatives**

Now it is the time to introduce the heavy artillery. In practice, you won't be computing the limit

\[f'(x_0) = \displaystyle\lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}\]very often. It is very important to know how to do it that way, but most of the times it won't be necessary.

**Example:** Compute the derivative of the function \(f(x)=x^3+x^2\).

**Solution:** What do we do here, do we apply the limit to compute the derivative?? Well, your line of reasoning should be the following: The function $f(x) = x^3+x^2$ corresponds to the sum of $x^3$ and $x^2$. The intuition is that if I could compute the derivative of each term \textit{separately}, then I could simplify the calculation.

In other words, if I knew what is the derivative of $x^3$, and if I also knew what is the derivative of $x^2$, then I should know what is the derivative of $x^3+x^2$.....

$\star$ __In fact, you do__. We have the following theorem:

**Theorem:** *The Derivative of the Sum of Two Functions*

Assume that $f(x)$ and $g(x)$ are \textit{differentiable} at $x_0$ (that means that the derivative exists at that point). Then, we have that

\[ \frac{d}{dx}(f(x)+g(x)) = \frac{df}{dx}(x) + \frac{dg}{dx}(x) \]In other words, the derivative of the sum is the sum of the derivatives (These are not empty words, they really describe the result accurately).
This is usually referred as the *Linearity Property of the derivative*

Now we show a result that will help us to compute a lot of derivatives:

**Theorem:** The following holds true for all $n\ne 0$:

**Proof:** We won't do anything too deep, just to save you from fatal boredom, but let's just do this one to get a feel of it. By definition

So, let's come back to the problem of finding the derivative of $f(x) = x^3+x^2$. Using the Linearity of derivatives we find that

\[\frac{d}{dx}(x^3+x^2) = \frac{d}{dx} x^3 + \frac{d}{dx}x^2 = 3x^2 + 2x\]Let's recall that $\frac{d}{dx}x^n = nx^{n-1}$, so applying that to the case $n=3$ and $n=2$ respectively we get the previous result. The Linearity property can be written in a more general way:

**Theorem:** Assume that $f(x)$ and $g(x)$ are \textit{differentiable} at $x_0$ and $a$ and $b$ are constants. Then

Below we show an example of how to apply this result:

**Example:** Compute the derivative of the function $f(x)=3 x^3+2 x^{1/2}$.

**Solution:** Using Linearity, we get that

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