The Sum Rule: Differentiation Made Easy

📜 Formula of the Sum Rule

If a function $f(x)$ is the sum of two functions $u(x)$ and $v(x)$, the formula is:

💡Good to know: The rule also applies to subtraction. In that case, it is called the difference rule: If $f(x) = u(x) - v(x)$, then the derivative is $f'(x) = u'(x) - v'(x)$.

🪜 Application in 3 Simple Steps

You can always apply the sum rule when the terms in your function are separated by a plus or minus sign.

1. Identify the summands: Break the function down into its individual parts (the summands) separated by $+$ or $-$.
2. Differentiate each summand individually: Apply the known differentiation rules to each part (e.g., the power rule or the derivative of constants).
3. Combine the results: Add (or subtract) the individual derivatives to obtain the total derivative of the function.

✏️ Examples for Illustration

The rule is best understood through practical examples.

Example 1: A Simple Polynomial Function

Given the function: $f(x) = x^4 + x^2$

• 1. Identify: $u(x) = x^4$ and $v(x) = x^2$.
• 2. Differentiate individually: The derivative of $u(x)$ is $u'(x) = 4x^3$. The derivative of $v(x)$ is $v'(x) = 2x$.
• 3. Combine: $f'(x) = u'(x) + v'(x) = 4x^3 + 2x$.

So the result is: $f'(x) = 4x^3 + 2x$

Example 2: Multiple Terms and Constants

Given the function: $f(x) = 2x^3 - 5x^2 + 7x - 3$

Here we differentiate each term individually:

• The derivative of $2x^3$ is $6x^2$.
• The derivative of $-5x^2$ is $-10x$.
• The derivative of $7x$ is $7$.
• The derivative of the constant $-3$ is $0$.

Combined, this gives the derivative: $f'(x) = 6x^2 - 10x + 7$

Example 3: Combination with Trigonometric Functions

Given the function: $f(x) = \sin(x) + x^3$

• 1. Identify: $u(x) = \sin(x)$ and $v(x) = x^3$.
• 2. Differentiate individually: The derivative of $u(x)$ is $u'(x) = \cos(x)$. The derivative of $v(x)$ is $v'(x) = 3x^2$.
• 3. Combine: $f'(x) = \cos(x) + 3x^2$.

So the result is: $f'(x) = \cos(x) + 3x^2$