Product & Chain Rule

The Product Rule states that if you have two functions u(x) and v(x), the derivative of their product is given by: \( (uv)' = u'v + uv' \).

The Chain Rule is used to differentiate composite functions. If \( y = f(g(x)) \), then the derivative is given by: \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Using the Product Rule, the derivative is: \( y' = (2)(5x - 4) + (2x + 1)(5) = 20x - 8 + 10x + 5 = 20x - 3 \).

Using the Product Rule, the derivative is: \( y' = 2x^2(3) + (2)(2x)(3x - 4) = 6x^2 - 16x \).

Using the Product Rule, the derivative is: \( y' = (2x)(x + 7) + (x^2 - 1)(1) = 2x^2 + 14x + x^2 - 1 = 3x^2 + 14x - 1 \).

Using the Chain Rule, the derivative is: \( y' = cos(2x^2) \cdot (4x) = 4x cos(2x^2) \).

Using the Product Rule, the derivative is: \( y' = (3x^4 - 7)(10x) + (12x^3)(5x^2 + 1) \).

If \( c \) is a constant and \( f(x) \) is a function, then \( \frac{d}{dx}[cf(x)] = c \cdot f'(x) \).

If \( f(x) \) and \( g(x) \) are functions, then \( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) \).

The derivative is: \( y' = 3x^2 + 10x - 2 \).