# How do you find the equation of the tangent line to graph #y=x^(1/x)# at the point #(1, 1)#?

The tangent line at

To do this, we will use logarithmic differentiation. That is, take the natural logarithm of both sides of the equation before taking the derivative:

Take the derivative of both sides of the equation. The left-hand side will require the chain rule. The right-hand side will use the product rule:

Solving for the derivative:

The slope of the tangent line is:

Or:

graph{(y-x^(1/x))(y-x)=0 [-1.628, 3.846, -0.475, 2.26]}

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To find the equation of the tangent line to the graph of y=x^(1/x) at the point (1, 1), we can use the concept of differentiation.

First, we need to find the derivative of the function y=x^(1/x). Using the chain rule, the derivative is given by:

dy/dx = (1/x) * (d/dx)(x^(1/x)) + x^((1/x)-1) * (d/dx)(1/x)

Simplifying this expression, we get:

dy/dx = (1/x) * (1/x)^(1/x) * (1 - ln(x)) - x^((1/x)-2) * (1/x^2)

Now, we substitute x=1 into the derivative to find the slope of the tangent line at the point (1, 1):

dy/dx = (1/1) * (1/1)^(1/1) * (1 - ln(1)) - 1^((1/1)-2) * (1/1^2) dy/dx = 1 * 1 * (1 - 0) - 1^(-1) * 1 dy/dx = 1 - 1 dy/dx = 0

Since the slope of the tangent line is 0, the equation of the tangent line is simply y = 1.

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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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