If g(x) = x , then g'(x) = 7x 6 7-6 Step 2 With A(x) = 1x1, f'(x) = -, g(x) = x", and g'(x) = 7x , we are now ready to substitute these terms into the quotient rule to find _ for y = dx x / Once the substitutions are made, continue to simplify. d f ( x ) F'(x)g(x) - f(x)g'(x) dx g(x) [g(x) 12 d Ixl (x7) - 1x1( ) dx X [x712 x' |x] - 716 x 14 X x14 6 x To conclude, if y then dy dxTutorial Exercise Calculate dx . Simplify your answer. HINT [See Examples 1 and 2.] Ixl X7 Step 1 In the given equation y - we have the quotient of two differentiable functions of x: (x] and x . Therefore, to differentiate we must first recall the Quotient Rule which states that if f and g are differentiable functions of x, then so is their quotient -, and the following formula applies, p f'(x)g(x) - 1(x)g'(x) dx g (x ) g( x ) 12 Many like to remember that the derivative of a quotient is the derivative of the top function times the bottom function minus the top function times the derivative of the bottom function, all derivative of derivative of top bottom top bottom ' (x ) g(x ) - F( X) g ( x ) dx g(x ) [g(x)1- bottom squared Up, if we let f(x) = Ixl and g(x) = x , then y - - Ixl For the given equation (). In other words, we can define f(x) = [x| as the top function and g(x) = x as the bottom function. g(x Note that since y is the quotient of two differentiable functions, we can use the quotient rule to find the derivative of y with respect to x or dy As a precursor to using the quotient rule, complete the following statements. If f(x) = Ixl, then f"(x) = If g(x) = x , then g'(x) - 7x Step 2 IX ! Ixl 7. Once the substitutions are made, continue to with F( x ) = Ixl , f ' ( x ) , g(x) = x", and g'(x) = 7x , we are now ready to substitute these terms into the quotient rule to find _ for y = dx X7 d F(x) F (x)9 (x) ( x)g ( x )