What are the basic rules for differentiating functions?

Basic Rules for Differentiating Functions

When you're finding the derivative (differentiating) of functions, these are the key rules to remember:

• Constant Rule:

The derivative of a constant is 0.
Example: If f(x) = 7, then f'(x) = 0.

• Power Rule:

For f(x) = x^n, the derivative is f'(x) = n x^(n-1).
Example: If f(x) = x^3, then f'(x) = 3x^2.

• Constant Multiple Rule:

If a constant multiplies a function, bring the constant outside the derivative: (a f(x))' = a f'(x).
Example: f(x) = 5x^2 ⇒ f'(x) = 5 2x = 10x.

• Sum Rule:

The derivative of a sum is the sum of the derivatives: (f(x) + g(x))' = f'(x) + g'(x).
Example: f(x) = x^2 + 3x ⇒ f'(x) = 2x + 3.

• Product Rule:

For multiplied functions: (f(x) g(x))' = f'(x)g(x) + f(x)g'(x).

• Quotient Rule:

For division: (f(x)/g(x))' = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2.

• Chain Rule:

For composite functions: If y = f(g(x)), then dy/dx = f'(g(x)) g'(x).

Takeaways:

• Use these rules to differentiate almost any common function.
• Practice by applying each rule to simple examples for mastery!