# What is the equation of the tangent line of #f(x)=ln(3x)^2-x^2# at #x=3#?

We have

The gradient of the tangent at any particular point is given by the derivative at that point.

Differentiating wrt

# \ \ \ \ \ f'(x) = 2ln(3x)*1/(3x)*3-2x #

# :. f'(x) = (2ln(3x))/x - 2x #

We need to find

#x=3 => f(3) \ \ = ln^2 9-9#

#x=3 => f'(3) = 2/3ln9-6#

So the tangent we seek passes through the point

# y - (ln^2 9-9) = (2/3ln9-6)(x - 3) #

# :. y - (ln^2 9-9) = (2/3ln9-6)(x - 3) #

# :. y - ln^2 9 + 9 = (2/3ln9-6)x - 3(2/3ln9-6) #

# :. y - ln^2 9 + 9 = (2/3ln9-6)x - 2ln9+18 #

# :. y = (2/3ln9-6)x +ln^2 9 - 2ln9+9 #

We can confirm this solution is correct by looking at the graph:

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To find the equation of the tangent line of f(x)=ln(3x)^2-x^2 at x=3, we need to find the slope of the tangent line and a point on the line.

First, we find the derivative of f(x) using the chain rule and power rule:

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

Next, we substitute x=3 into f'(x) to find the slope of the tangent line:

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

Finally, we evaluate f'(3) to find the slope:

f'(3) = 2(ln(9))(1/9)(3) - 6

Therefore, the slope of the tangent line at x=3 is f'(3).

To find a point on the line, we substitute x=3 into f(x):

f(3) = ln(3(3))^2 - 3^2

Finally, we evaluate f(3) to find the y-coordinate of the point:

f(3) = ln(9) - 9

Therefore, a point on the tangent line is (3, ln(9) - 9).

Using the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept, we can write the equation of the tangent line:

y = f'(3)(x - 3) + f(3)

Substituting the values we found earlier, the equation of the tangent line is:

y = (2(ln(9))(1/9)(3) - 6)(x - 3) + (ln(9) - 9)

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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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