If \(y = f(x) · g(x)\), then:
\[ y' = f' g + f g' \]Memory trick: "first · derivative of second + second · derivative of first"
Example: \(y = x^{2} · \sin(x)\)
\(y' = (2x) \sin(x) + x^{2} \cos(x)\)
If \(y = f(x) / g(x)\), then:
\[ y' = \frac{f' g - f g'}{g^2} \]Memory trick: "low d-high minus high d-low, over low squared"
Example: \(y = (x^2 + 1) / (x - 3)\)
\(y' = [(2x)(x-3) - (x^2+1)(1)] / (x-3)^2\)
Used when one function is inside another: \(y = f(g(x))\)
\[ \frac{dy}{dx} = f'(g(x)) · g'(x) \]Example: \(y = (3x + 1)^4\)
Let \(u = 3x + 1 → y = u^4 → dy/dx = 4u^3 · 3 = 12(3x + 1)^3\)
Memory trick: "derivative of outside, times derivative of inside"