How do you differentiate #f(x)= ( x +2sinx )/ (x + 4 )# using the quotient rule?

Question

How do you differentiate #f(x)=  (  x   +2sinx    )/  (x  +  4   )# using the quotient rule?

Scarlett Allison · Accepted Answer

#f'(x)=(2(cosx(x+4)-sinx+2))/(x+4)^2#

According to the quotient rule, a function's derivative can be expressed as the quotient of two other functions.

#f=(g/h)#

possesses a derivative of

#f^'=(g^'h-gh^')/h^2#

For #f(x)=(x+2sinx)/(x+4)#:

We see that #g=x+2sinx# so #g^'=1+2cosx# and #h=x+4# so #h^'=1#.

This results in:

#f'(x)=((1+2cosx)(x+4)-(x+2sinx)(1))/(x+4)^2#

#f'(x)=(x+4+2xcosx+8cosx-x-2sinx)/(x+4)^2#

#f'(x)=(2(cosx(x+4)-sinx+2))/(x+4)^2#

Evelyn Agard · Answer

undefined To differentiate $ f(x) = \frac{x + 2\sin(x)}{x + 4} $ using the quotient rule, follow these steps:

1. Identify $ u(x) $ and $ v(x) $.
   $ u(x) = x + 2\sin(x) $ and $ v(x) = x + 4 $.

2. Compute $ u'(x) $ and $ v'(x) $.
   $ u'(x) = 1 + 2\cos(x) $ and $ v'(x) = 1 $.

3. Apply the quotient rule formula:
   $$ f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} $$

4. Substitute the values:
   $$ f'(x) = \frac{(x + 4)(1 + 2\cos(x)) - (x + 2\sin(x))(1)}{(x + 4)^2} $$

5. Simplify the expression:
   $$ f'(x) = \frac{x + 4 + 2x\cos(x) + 4\cos(x) - x - 2\sin(x)}{(x + 4)^2} $$

6. Further simplify if needed:
   $$ f'(x) = \frac{2x\cos(x) + 4\cos(x) - 2\sin(x) + 4}{(x + 4)^2} $$