DIFFERENTIATION 05

Sum, Difference and Constant Multiple Rules

Functions made from simpler terms can often be differentiated one term at a time.

The main idea

Suppose

\[ y=f(x)+g(x). \]

Then

\[ \boxed{ \frac{dy}{dx} = f'(x)+g'(x) } \]

Differentiation can be applied separately to each term in a sum.

1. The sum rule

The derivative of a sum is the sum of the derivatives:

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

For example,

\[ y=x^3+x^2. \]

Differentiate each term:

\[ \frac{dy}{dx} = 3x^2+2x. \]

2. Why the sum rule works

From first principles,

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

\[ \lim_{h\to0} \frac{ [f(x+h)+g(x+h)] - [f(x)+g(x)] }{h}. \]

Rearranging the numerator,

\[ = \lim_{h\to0} \left[ \frac{f(x+h)-f(x)}{h} + \frac{g(x+h)-g(x)}{h} \right]. \]

Taking the limits separately gives

\[ f'(x)+g'(x). \]

So the rule follows directly from the definition of the derivative.

3. The difference rule

Subtraction behaves in the same way:

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

For example,

\[ y=x^4-x^2. \]

Then

\[ \frac{dy}{dx} = 4x^3-2x. \]

4. The constant multiple rule

Suppose a function is multiplied by a constant \(c\):

\[ y=cf(x). \]

Then

\[ \boxed{ \frac{d}{dx}[cf(x)] = c f'(x) } \]

The constant remains unchanged while the function is differentiated.

5. Example

Consider

\[ y=5x^3. \]

Since

\[ \frac{d}{dx}x^3=3x^2, \]

we obtain

\[ \frac{dy}{dx} = 5(3x^2) = 15x^2. \]

6. Combining the rules

Consider

\[ y=4x^5-3x^3+2x^2-7x+9. \]

Differentiate each term:

\[ \frac{dy}{dx} = 20x^4-9x^2+4x-7. \]

The constant \(9\) has derivative \(0\).

7. Why constants disappear

Suppose

\[ f(x)=7. \]

No matter how \(x\) changes, the output remains \(7\).

Therefore its rate of change is zero:

\[ \boxed{ \frac{d}{dx}(7)=0 } \]

More generally,

\[ \frac{d}{dx}(c)=0. \]

8. Example with negative powers

Consider

\[ y=3x^2+4x^{-1}. \]

Differentiate term by term:

\[ \frac{dy}{dx} = 6x-4x^{-2}. \]

Equivalently,

\[ \boxed{ \frac{dy}{dx} = 6x-\frac{4}{x^2} } \]

9. Example with roots

Consider

\[ y=2\sqrt{x}+x^3. \]

First write the root as a power:

\[ y=2x^{1/2}+x^3. \]

Then

\[ \frac{dy}{dx} = 2\left(\frac12x^{-1/2}\right) + 3x^2. \]

Therefore,

\[ \boxed{ \frac{dy}{dx} = \frac{1}{\sqrt{x}} + 3x^2 } \]

for \(x>0\).

10. What these rules allow us to do

A polynomial may contain many terms, but each term can be handled separately.

\[ \boxed{ \text{differentiate each term} \quad\longrightarrow\quad \text{combine the results} } \]

For example,

\[ 3x^4+2x^3-5x+8 \]

becomes

\[ 12x^3+6x^2-5. \]

11. But multiplication of functions is different

Suppose

\[ y=f(x)g(x). \]

We cannot generally differentiate both functions separately and simply multiply:

\[ \frac{d}{dx}[f(x)g(x)] \neq f'(x)g'(x). \]

For example,

\[ x^2\cdot x^3=x^5. \]

Its derivative is

\[ 5x^4. \]

But differentiating the two factors separately and multiplying would give

\[ (2x)(3x^2)=6x^3, \]

which is incorrect.

Products therefore require a new rule.

12. The rules so far

\[ \frac{d}{dx}(c)=0 \] \[ \frac{d}{dx}(x^n)=nx^{n-1} \] \[ \frac{d}{dx}[f+g] = f'+g' \] \[ \frac{d}{dx}[f-g] = f'-g' \] \[ \frac{d}{dx}[cf] = cf' \]

These rules are enough to differentiate any polynomial directly.