\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (x) = 1} \end{align*}\]

Let’s learn how to find the derivative of the function $f(x) = x$. It’s simple, we will show every step.

What is a Derivative?

A derivative tells us how a function changes as the input $x$ changes. Think of it like finding out how fast or slow something is moving. In math, it helps us understand the rate at which $y$ changes with respect to $x$.

Step-by-Step Explanation

Step 1: Write the Function

Our function is:
\[\begin{align*} f(x) = x \end{align*}\]

Step 2: Identify the Exponent of $x$

Remember that $x$ is the same as $x^1$ because any number raised to the power of 1 is itself:
\[\begin{align*} f(x) = x^1 \end{align*}\]

Step 3: Recall the Power Rule

The Power Rule is a basic rule in calculus for finding derivatives. It states:
\[\begin{align*} \frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1} \end{align*}\]
where $n$ is the exponent.

Step 4: Apply the Power Rule

Let’s apply the Power Rule to our function:

• The exponent $n$ is 1.
• Plugging into the Power Rule:
  \[\begin{align*} \frac{d}{dx} (x^1) = 1 \cdot x^{1-1} \end{align*}\]

Step 5: Simplify

Simplify the expression:

• Subtract $1 – 1$ in the exponent:
  \[\begin{align*} x^{1-1} = x^0 \end{align*}\]
• Any number raised to the power of 0 is 1:
  \[\begin{align*} x^0 = 1 \end{align*}\]
• Thus,
  \[\begin{align*} \boxed{ \frac{d}{dx} (x^1) = 1 \cdot 1 = 1 } \end{align*}\]

Conclusion

We have learned that the derivative of $x$ is 1. This means that for every increase of 1 in $x$, the function $f(x) = x$ also increases by 1. Remember, the derivative tells us the rate of change or in other words the slope of the function.

\[\begin{align*} \boxed{\displaystyle \frac{d}{dx} (x) = 1} \end{align*}\]

Sample Problem

Now, let’s try a similar problem to practice what we’ve learned.

Problem: Find the Derivative of $g(x) = 4x$

Step 1: Identify the Constant and the Variable

Here, $4$ is a constant, and $x$ is our variable raised to the power of 1:
\[\begin{align*} g(x) = 4 \cdot x^1 \end{align*}\]

Step 2: Recall the Constant Multiple Rule

The Constant Multiple Rule states:
\[\begin{align*} \frac{d}{dx} [c \cdot f(x)] = c \cdot \frac{d}{dx} [f(x)] \end{align*}\]
where $c$ is a constant.

Step 3: Apply the Constant Multiple Rule and Power Rule

First, find the derivative of $x^1$ using the Power Rule:
\[\begin{align*} \frac{d}{dx} (x^1) = 1 \cdot x^{1-1} = 1 \cdot x^0 = 1 \end{align*}\]

Now, multiply by the constant $4$:
\[\begin{align*} \frac{d}{dx} [4x] = 4 \cdot \frac{d}{dx} [x] = 4 \cdot 1 = 4 \end{align*}\]

The derivative of $g(x) = 4x$ is:
\[\begin{align*} \boxed{\frac{d}{dx} (4x) = 4} \end{align*}\]