# What is the equation of the line that is normal to #f(x)= x^3 ln(x^2+1)-x/(x^2+1) #at # x= 1 #?

Rewriting:

Differentiating with the product rule in both cases:

The chain rule will apply twice:

Which, if we like, can be "simplified" as:

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To find the equation of the line that is normal to the function f(x) = x^3 ln(x^2+1) - x/(x^2+1) at x = 1, we need to determine the slope of the normal line and the point of tangency.

First, we find the derivative of f(x) using the product rule and quotient rule:

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

Simplifying further:

f'(x) = 3x^2 ln(x^2+1) + 2x^4 / (x^2+1) - (x^2+1 - 2x^2) / (x^2+1)^2

Next, we substitute x = 1 into f'(x) to find the slope of the tangent line at x = 1:

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

Simplifying further:

f'(1) = 3 ln(2) + 2/2 - (2 - 2) / 4

f'(1) = 3 ln(2) + 1/2

Therefore, the slope of the tangent line at x = 1 is 3 ln(2) + 1/2.

Since the normal line is perpendicular to the tangent line, the slope of the normal line is the negative reciprocal of the slope of the tangent line.

Hence, the slope of the normal line is -1 / (3 ln(2) + 1/2).

Using the point-slope form of a line, we can write the equation of the normal line:

y - f(1) = (-1 / (3 ln(2) + 1/2))(x - 1)

Simplifying further:

y - f(1) = (-1 / (3 ln(2) + 1/2))(x - 1)

Therefore, the equation of the line that is normal to f(x) = x^3 ln(x^2+1) - x/(x^2+1) at x = 1 is given by the above equation.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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