Calculating Derivatives: Constants and Sums

The Power Rule can be used to find the derivative of functions such as $f(x) = x^n$. However, how can one find the derivatives of functions that are simpler, like $f(x) = 5$? Or functions that are combinations of terms, like $f(x) = x^2 + x$?

We need a couple more rules to handle these situations. Fortunately, these rules are quite intuitive and follow directly from the idea of the derivative as a rate of change or the slope of a tangent line. In this section, we'll cover how to find the derivatives of constants and sums (or differences) of functions.

The Derivative of a Constant: No Change Means Zero Slope

Let's start with the simplest type of function: a constant function. Consider the function:

$f(x) = 5$

This function always outputs the value 5, no matter what input value $x$ you give it. What does its graph look like? It's a horizontal line passing through 5 on the y-axis.

The graph of a constant function $f(x)=c$ is a horizontal line.

Remember that the derivative represents the slope of the tangent line to the function's graph. What's the slope of a horizontal line? It's zero. The line isn't rising or falling; its rate of change is zero everywhere.

This gives us our first rule:

The Constant Rule: If $c$ is a constant number, then the derivative of the function $f(x) = c$ is zero.

Using derivative notation:

If $f(x) = c$, then $f'(x) = 0$.

Alternatively, using Leibniz notation:

$\frac{d}{dx}(c) = 0$

This makes sense intuitively: if a quantity never changes (it's constant), its rate of change is zero.

Examples:

• If $f(x) = 100$, then $f'(x) = 0$.
• If $y = -3.14$, then $\frac{dy}{dx} = 0$.
• $\frac{d}{dx}(2) = 0$.

Combining Functions: The Sum and Difference Rules

Now, what if we have a function formed by adding or subtracting simpler functions? For instance, consider:

$h(x) = x^2 + x$

We know how to find the derivative of $x^2$ (it's $2x$ by the Power Rule) and the derivative of $x$ (which is $x^1$, so its derivative is $1x^0 = 1$, also by the Power Rule). How does the addition affect the derivative?

Calculus provides a straightforward rule: the derivative of a sum of functions is simply the sum of their individual derivatives. The same applies to differences.

The Sum Rule: If $h(x) = f(x) + g(x)$, then $h'(x) = f'(x) + g'(x)$.

In Leibniz notation:

$\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$

The Difference Rule: If $h(x) = f(x) - g(x)$, then $h'(x) = f'(x) - g'(x)$.

In Leibniz notation:

$\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}f(x) - \frac{d}{dx}g(x)$

Essentially, you can differentiate a function term by term.

Let's apply this to our example $h(x) = x^2 + x$:

1. Find the derivative of the first term, $x^2$. Using the Power Rule, $\frac{d}{dx}(x^2) = 2x$.
2. Find the derivative of the second term, $x$. Using the Power Rule, $\frac{d}{dx}(x) = 1$.
3. Add the results: $h'(x) = 2x + 1$.

Handling Constant Multiples

What about a function like $f(x) = 3x^2$? Here, $x^2$ is multiplied by a constant, 3. There's a simple rule for this too. Constant factors just "tag along" when you differentiate.

The Constant Multiple Rule: If $h(x) = c \cdot f(x)$, where $c$ is a constant, then $h'(x) = c \cdot f'(x)$.

In Leibniz notation:

$\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}f(x)$

So, for $f(x) = 3x^2$:

1. Identify the constant multiple: $c=3$.
2. Identify the function being multiplied: $f(x) = x^2$.
3. Find the derivative of that function: $\frac{d}{dx}(x^2) = 2x$.
4. Multiply the result by the constant: $f'(x) = 3 \cdot (2x) = 6x$.

Putting It All Together

Now we can combine the Constant Rule, Sum/Difference Rule, Constant Multiple Rule, and Power Rule to differentiate any polynomial function. A polynomial is just a sum of terms where each term is a constant multiplied by $x$ raised to a non-negative integer power (like $5x^3 - 2x + 7$).

Let's find the derivative of $p(x) = 4x^3 - 5x^2 + x - 2$.

We can differentiate this term by term:

1. Term 1: $4x^3$
   • Derivative of $x^3$ is $3x^2$.
   • Apply Constant Multiple Rule: $\frac{d}{dx}(4x^3) = 4 \cdot (3x^2) = 12x^2$.
2. Term 2: $-5x^2$
   • Derivative of $x^2$ is $2x$.
   • Apply Constant Multiple Rule: $\frac{d}{dx}(-5x^2) = -5 \cdot (2x) = -10x$.
3. Term 3: $+x$ (or $+1x^1$)
   • Derivative of $x^1$ is $1x^0 = 1$.
   • Apply Constant Multiple Rule (optional here): $\frac{d}{dx}(1x) = 1 \cdot (1) = 1$.
4. Term 4: $-2$
   • This is a constant.
   • Apply Constant Rule: $\frac{d}{dx}(-2) = 0$.

Now, combine the derivatives of each term using the Sum/Difference Rule:

$p'(x) = 12x^2 - 10x + 1 - 0$

$p'(x) = 12x^2 - 10x + 1$

With these rules (Constant, Constant Multiple, Sum/Difference, and Power), you now have the tools to find the derivative of any polynomial function. These form the basic toolkit for differentiation, which we'll use extensively when we start looking at how to optimize functions, a core task in training machine learning models.