# What is the equation of the normal line of #f(x)=sqrt(x/(x+1) # at #x=4 #?

Now, to diff. y, we can use the Quotient Rule. Instead, have a look at these two methods :-

METHOD I :-

METHOD II :-

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To find the equation of the normal line of f(x) = sqrt(x/(x+1) at x = 4, we need to determine the slope of the tangent line at x = 4 and then find the negative reciprocal of that slope to obtain the slope of the normal line.

To find the slope of the tangent line, we can take the derivative of f(x) with respect to x and evaluate it at x = 4.

The derivative of f(x) = sqrt(x/(x+1) can be found using the quotient rule:

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

Simplifying this expression, we get:

f'(x) = (x+1 - x) / (2 * (x+1)^(3/2))

f'(x) = 1 / (2 * (x+1)^(3/2))

Now, we can evaluate f'(x) at x = 4:

f'(4) = 1 / (2 * (4+1)^(3/2))

f'(4) = 1 / (2 * 5^(3/2))

f'(4) = 1 / (2 * 5 * sqrt(5))

f'(4) = 1 / (10 * sqrt(5))

The slope of the tangent line at x = 4 is 1 / (10 * sqrt(5)).

To find the slope of the normal line, we take the negative reciprocal of the slope of the tangent line:

Slope of normal line = -1 / (1 / (10 * sqrt(5)))

Slope of normal line = -10 * sqrt(5)

Now, we have the slope of the normal line. To find the equation of the normal line, we use the point-slope form of a line and substitute the values of x = 4 and the slope:

y - f(4) = (-10 * sqrt(5)) * (x - 4)

Simplifying further, we get:

y - sqrt(4/(4+1)) = (-10 * sqrt(5)) * (x - 4)

y - sqrt(4/5) = (-10 * sqrt(5)) * (x - 4)

This is the equation of the normal line of f(x) = sqrt(x/(x+1)) at x = 4.

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