Calculus:More Differentation Rules

1. REDIRECT Calculus/More Differentiation Rules

The product rule

The product of two differentiable funcations f and g is itself differentiable. Moreover, the derivative of fg is the first function times the derivative of the 2nd plus the 2nd fuction times the derivative of the 1st

y' of y = u x v u & v representing different fuctions like --> (x-3)(x+7)

y' = uv' + u'v

Example 1: Find y' of [math]\displaystyle{ y=(2x-7)(7x^2+8) }[/math]

    remember: y' = uv' + u'v

              [math]\displaystyle{ y' = (2x-7)[(7*2)x] + (2)(7x^2+8) }[/math] product rule
              [math]\displaystyle{    = (2x-7)(14x) + 2(7x^2+8)  }[/math] simplify
              [math]\displaystyle{    = 42x^2-98x+16  }[/math]

The quotient rule

Similar to product rule,

y' of y = u / v u & v representing different fuctions like --> (x-3)(x+7)

y' = (u'v - uv') / v^2

Example 1: Find y' of [math]\displaystyle{ y=(3x)(2x^2) }[/math]

              [math]\displaystyle{ y' = [(3)(2x^2) - (3x)(4x)] / (2x^2)^2  }[/math] quotient rule
              [math]\displaystyle{    = (6x^2-12x^2) / (4x^4)  }[/math] simplify
              <math>   = (-6x^2) / (4x^4)
                       = (-3)/(2x^2)