Difference Rule

Definition

The Difference Rule states that the derivative of a difference of two functions is the difference of their derivatives. Mathematically, if $f(x)$ and $g(x)$ are differentiable, then $(f-g)' = f' - g'$.

5 Must Know Facts For Your Next Test

1. The Difference Rule applies to any pair of differentiable functions.
2. It simplifies the process of finding derivatives when dealing with subtraction.
3. The rule can be used in conjunction with other differentiation rules like the Product and Quotient Rules.
4. You can apply this rule as $(f-g)'(x) = f'(x) - g'(x)$ for all $x$ in the domain where both functions are differentiable.
5. This rule is often introduced alongside the Sum Rule, which deals with addition instead of subtraction.

Review Questions

• What is the derivative of $h(x) = f(x) - g(x)$ using the Difference Rule?
• If $f(x) = x^2$ and $g(x) = \sin(x)$, what is $(f-g)'(x)?$
• How does the Difference Rule simplify finding the derivative of a function defined by subtraction?

Related terms

Sum Rule:

The Sum Rule states that the derivative of a sum of two functions is the sum of their derivatives: $(f+g)' = f' + g'.$

Product Rule: The Product Rule states that the derivative of a product of two functions is given by $(fg)' = f'g + fg'.$

Quotient Rule: The Quotient Rule states that the derivative of a quotient is given by $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$, provided that $g \neq 0$.