2.3 The Product Rule Unit 2: Derivatives MCV 4U1: Calculus and Vectors Investigate Find the derivative of the following function listed in two equivalent forms: ( ) ( ) ( )( ) Product Rule If f(x) and g(x) are differentiable at x and p(x) = f(x)g(x) then: ( ) ( ) ( ) [ ( ) ( )] ( ( ) ( ) ( ) ( ) ( ) ( ) ) Example: Differentiate using the product rule for the function ( ) ( Example: Find the equation of the tangent line to Example: Consider the function ( ) a) Differentiate and simplify √ ( √ )( ( )( ). ) at the point P (1, 4). ) b) Find the values of x where the function is not differentiable. Explain. c) Find the values of x where the tangent line is horizontal. Product of Three Functions If f(x), g(x) and h(x) are differentiable at x and p(x) = f(x)g(x)h(x) then: ( ) ( ) ( ) ( ) ( ) ( ) ( ) Example: Differentiate but do not simplify ( ) Power of a Function Rule (Chain Rule) . If ( ) If f(x) is differentiable at x, then so is ( ) ( [ ( )] )( [ ( )] then, ( ) ) ( Example: Differentiate Example: Given ( ) (√ ( ) ( ) ( ) ( Page 90-93 #1, 2, 5alt, 6, 7, 8a, 13 ) ) , find the velocity at t = 1s. )( )