# How do you find the equation of the line tangent to the graph of #(2x+3)^(1/2)# at the point x=3?

First use the chain rule to get the derivative:

Let:

The equation of the line is of the form:

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To find the equation of the line tangent to the graph of (2x+3)^(1/2) at the point x=3, we need to find the derivative of the function and evaluate it at x=3.

First, we find the derivative of (2x+3)^(1/2) using the power rule of differentiation. The derivative is given by:

dy/dx = (1/2)(2x+3)^(-1/2)(2)

Simplifying this expression, we get:

dy/dx = (2)/(2x+3)^(1/2)

Next, we evaluate the derivative at x=3:

dy/dx = (2)/(2(3)+3)^(1/2) = (2)/(6+3)^(1/2) = (2)/(9)^(1/2) = (2)/3

So, the slope of the tangent line at x=3 is 2/3.

Now, we have the slope of the tangent line and the point (3, (2(3)+3)^(1/2)) on the graph. We can use the point-slope form of a line to find the equation of the tangent line:

y - y1 = m(x - x1)

Plugging in the values, we get:

y - (2(3)+3)^(1/2) = (2/3)(x - 3)

Simplifying further:

y - (3)^(1/2) = (2/3)(x - 3)

This is the equation of the line tangent to the graph of (2x+3)^(1/2) at the point x=3.

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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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