Let f(x) = [(√x)-7]/[(√x)+7]. What is f'(x)?

What is the easiest way to find the derivative of this? Should I remove all the radicals and use quotient rule, like

f'(x)= ((x^0.5) + 7)(0.5x^-0.5) - ((x^0.5)-7)(0.5x^-0.5) / ((x^0.5) + 7)^2

Is this right? How do I simplify this? please show me thanks

Radicals or exponents. Tomato, tomahto. Either way, it's slightly messy, but simplifies rather nicely.

f = (√x-7)/(√x+7)
As you say,
f' = (1/2√x)(√x+7)-(√x-7)(1/2√x))/(√x+7)^2
= (7/√x)/(√x+7)^2
= 7/[√x(√x+7)^2)

thank you

To find the derivative of f(x), you can simplify the expression before applying the quotient rule. Here's the step-by-step process:

Step 1: Simplify the expression.
Start by multiplying the numerator and denominator of f(x) by the conjugate of the denominator, (√x - 7):

f(x) = [(√x - 7)(√x - 7)] / [(√x + 7)(√x - 7)]
= (x - 14√x + 49) / (x - 49)

Step 2: Now, we can apply the quotient rule to find the derivative. The quotient rule states that for a function u/v, the derivative is given by:

[u'v - uv'] / v^2

Using this rule, we can find f'(x):

f'(x) = [(1)(x - 49) - (x - 14√x + 49)(1)] / (x - 49)^2
= (x - 49 - x + 14√x - 49) / (x - 49)^2
= 14√x / (x - 49)^2

So, the derivative f'(x) is equal to 14√x divided by (x - 49) squared.

Note: The expression you provided in the question is incorrect. The correct derivative after simplification is f'(x) = 14√x / (x - 49)^2.