computing a derivative - calculus question

Question

Compute the following derivatives. Assume a and b are constants, f and g are functions, and x is the input variable. Your answers may include any or all of the symbols x, a, b, f and g, as well as the derivatives of f and g.

y= x+f/g = __________________

Answer 1

y = x + f/g

derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared

derivative of x is 1

y' = 1 + (gf'- fg')g^2

y = (g^2 + gf' - fg')/g^2

or

rewrite the quotient as a product

y = x + (f)(g^-1)

derivative of a product is the 1st term times derivative of 2nd plus 2nd times derivative of 1st

y'(x) = 1 + (f)(-g^-2)(g') + (g^-1)f'

y' = 1 - fg'/g^2 + f'/g

y' = 1 - fg'/g^2 + f'g/g^2

dy/dx =

y' = (g^2 - fg' + f'g)/g^2

Answer 2

Assuming f and g are functions of x and are not just constants

1) Take the directives of x

d/dx[x^n]=n*x^(n-1)

The derivative of x is 1 in this case since x is raised to the zero power and then x is multiplied by the original exponent 1.

2) Now find the derivative of f/g.

To do this we need to use the quotient rules

d/dx[f(x)/g(x)]=[g(x)*f '(x)-f(x)*g '(x)] / (g(x))^2

Putting it all together

y'=1+[g*f '-f*g '] / (g)^2