Derivative Rules 🧮

Here are the fundamental derivative rules for calculus:

1. Power Rule

For $ f(x) = x^n $, the derivative is:
$ f'(x) = n \cdot x^{n-1} $
⚡ Example: $ \frac{d}{dx}(x^3) = 3x^2 $

2. Chain Rule

For composite functions $ f(g(x)) $:
$ f'(x) = f'(g(x)) \cdot g'(x) $
📚 Use case: Differentiating $ \sin(2x) $ or $ (5x+3)^4 $

3. Product Rule

For $ f(x) = u(x) \cdot v(x) $:
$ f'(x) = u'(x)v(x) + u(x)v'(x) $
🧮 Practice: $ \frac{d}{dx}(x^2 \cdot \cos x) $

4. Quotient Rule

For $ f(x) = \frac{u(x)}{v(x)} $:
$ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} $
🔍 Tip: Simplify before applying the rule if possible

5. Implicit Differentiation

Differentiate both sides with respect to $ x $:
$ \frac{d}{dx}(y) \cdot \frac{dy}{dx} = \frac{d}{dx}(F(x,y)) $
⚡ Useful for: Equations like $ x^2 + y^2 = 25 $

Need help applying these rules? Try our interactive Derivative Calculator to practice!