Constant Multiple and Sum Rules

You can pull constants out front when differentiating!

Examples of Constant Multiple Rule

Example 1: $\frac{d}{dx}[5x^3]$

Pull out the 5: $= 5 \cdot \frac{d}{dx}[x^3] = 5 \cdot 3x^2 = 15x^2$

Example 2: $\frac{d}{dx}[-2x^4]$

Pull out the -2: $= -2 \cdot \frac{d}{dx}[x^4] = -2 \cdot 4x^3 = -8x^3$

Example 3: $\frac{d}{dx}\left[\frac{3}{4}x^2\right]$

$= \frac{3}{4} \cdot \frac{d}{dx}[x^2] = \frac{3}{4} \cdot 2x = \frac{3x}{2}$

The Sum Rule

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

Differentiate term by term!

The Difference Rule

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

Same as the sum rule, but with subtraction!

Examples of Sum/Difference Rules

Example 1: $\frac{d}{dx}[x^3 + x^2]$

Differentiate each term: $= \frac{d}{dx}[x^3] + \frac{d}{dx}[x^2] = 3x^2 + 2x$

Example 2: $\frac{d}{dx}[x^5 - x^3 + x]$

$= \frac{d}{dx}[x^5] - \frac{d}{dx}[x^3] + \frac{d}{dx}[x]$ $= 5x^4 - 3x^2 + 1$

Example 3: $\frac{d}{dx}[2x^4 + 3x^2 - 7]$

$= 2 \cdot 4x^3 + 3 \cdot 2x - 0$ $= 8x^3 + 6x$

Combining All the Rules

For polynomials, we use all three rules together!

General polynomial: $\frac{d}{dx}[a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0]$

$= na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + ... + a_1$

Step-by-Step Process

Find the derivative of: $f(x) = 4x^5 - 3x^3 + 7x - 2$

Step 1: Differentiate each term separately

• $\frac{d}{dx}[4x^5] = 4 \cdot 5x^4 = 20x^4$
• $\frac{d}{dx}[-3x^3] = -3 \cdot 3x^2 = -9x^2$
• $\frac{d}{dx}[7x] = 7 \cdot 1 = 7$
• $\frac{d}{dx}[-2] = 0$

Step 2: Combine them $f'(x) = 20x^4 - 9x^2 + 7$

Why These Rules Work

Constant Multiple: Constants are just scaling factors. The rate of change gets scaled too!

Sum/Difference: Rates of change add/subtract independently. If one quantity increases at rate a and another at rate b, the total increases at rate a + b.

Extended to More Terms

These rules work for any number of terms:

$\frac{d}{dx}[f_1 + f_2 + f_3 + ... + f_n] = f_1' + f_2' + f_3' + ... + f_n'$

Important Note

These rules apply to sums and differences, NOT products or quotients!

❌ Wrong: $\frac{d}{dx}[f \cdot g] \neq f' \cdot g'$

We need special rules for products and quotients (coming next)!

Common Patterns

Practice Tips

1. Handle each term independently
2. Pull out constants first
3. Apply power rule to each term
4. Remember: derivative of constant = 0
5. Combine all the results

Second term: $\frac{d}{dx}[-5x]$ $= -5 \cdot 1 = -5$

Third term: $\frac{d}{dx}[7]$ $= 0$ (constant)

Combine: $f'(x) = 6x - 5$

Answer: f'(x) = 6x - 5