Constant Multiple and Sum Rules

Constant Multiple and Sum/Difference Rules

These rules let us differentiate polynomials and combinations of functions easily!

The Constant Multiple Rule

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

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

| Function | Derivative | |----------|------------| | $x^n$ | $nx^{n-1}$ | | $cx^n$ | $cnx^{n-1}$ | | $x^n + x^m$ | $nx^{n-1} + mx^{m-1}$ | | $ax^2 + bx + c$ | $2ax + b$ |

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