# How do you find an equation of the tangent line to the curve at the given point #y=tan^2(3x)# and #x=pi/4#?

The equation of the tangent at

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To find the equation of the tangent line to the curve y = tan^2(3x) at the point x = π/4, we need to find the derivative of the function and evaluate it at x = π/4.

First, let's find the derivative of y = tan^2(3x) using the chain rule.

dy/dx = 2tan(3x) * sec^2(3x) * 3

Now, substitute x = π/4 into the derivative expression to find the slope of the tangent line at that point.

dy/dx = 2tan(3(π/4)) * sec^2(3(π/4)) * 3

Evaluate the trigonometric functions:

dy/dx = 2tan(3π/4) * sec^2(3π/4) * 3

Simplify the trigonometric expressions:

dy/dx = 2(-1) * (2/√2)^2 * 3

dy/dx = -8 * 3

dy/dx = -24

The slope of the tangent line at x = π/4 is -24.

Now, we can use the point-slope form of a linear equation to find the equation of the tangent line.

y - y1 = m(x - x1)

Substitute the values of the point (x1, y1) = (π/4, tan^2(3π/4)) and the slope m = -24 into the equation:

y - tan^2(3π/4) = -24(x - π/4)

This is the equation of the tangent line to the curve y = tan^2(3x) at the point x = π/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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