For y=x/(x+4)^0.5, solve dy/dx
As x is present as both numerator and denominator (top & bottom), we're going to use the quotient rule to solve this. The quotient rule is as follows:For y=f(x)/g(x), dy/dx=(f'(x).g(x)-f(x).g'(x))/((g(x))^2) ---See whiteboard for a clearer demonstration
Considering that f'(x).g(x)= The first derivative of f(x) multiplied by g(x). First derivative simply means df(x)/dx i.e just differentiate f(x).
When applied to our question we can see that: f(x)=x so f'(x)=1 and g(x)=(x+4)^0.5 so g'(x)=0.5(x+4)^-0.5 following the chain rule.
We can now substitute these terms into the quotient rule so that dy/dx=((x+4)^0.5 -0.5x(x+4)^-0.5)/(x+4)
This can be further simplified by multiplying all terms by (x+4)^-0.5 to get dy/dx=(x+4-0.5x)/(x+4)^1.5=(0.5x+4)/((x+4)^1.5)
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