# How do you find the equation of the tangent line to the curve #y=ln(3x-5)# at the point where x=3?

The equation of the tangent line is composed of two parts: the derivative (slope), and the

First we find the derivative of our function at

However, the chain rule tells us that we need to multiply this by the derivative of the "inside" function (in our case,

To find the derivative at

Thus the slope of our tangent line is

We can finally move on to the last step - finding the actual equation. Because we know the tangent line is straight, it will be of the form

Let's plug these numbers into our equation:

That's our

See the graph below for a visual description of the solution.

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To find the equation of the tangent line to the curve y=ln(3x-5) at the point where x=3, we need to find the slope of the tangent line and the coordinates of the point of tangency.

First, we find the derivative of the function y=ln(3x-5) using the chain rule. The derivative is given by dy/dx = 1/(3x-5).

Next, we substitute x=3 into the derivative to find the slope of the tangent line at x=3. Substituting x=3 into the derivative, we get dy/dx = 1/(3(3)-5) = 1/4.

Now, we need to find the y-coordinate of the point of tangency. Substituting x=3 into the original function y=ln(3x-5), we get y=ln(3(3)-5) = ln(4).

Therefore, the point of tangency is (3, ln(4)).

Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute the values we found to get the equation of the tangent line.

Substituting m=1/4, x1=3, and y1=ln(4), we get y - ln(4) = (1/4)(x - 3).

Simplifying the equation, we have y = (1/4)x - 3/4 + ln(4).

Therefore, the equation of the tangent line to the curve y=ln(3x-5) at the point where x=3 is y = (1/4)x - 3/4 + ln(4).

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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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