# How do you find the derivative with respect to x of #f(x)=x^2# and use it to find the equation of the tangent line to #y=x^2# at x=2?

Use the power rule .

Plug in 2 into this to find the slope:

Now plug in x=2 into the first equation to get a point which will be used to find the equation.

Now we have the point (2,4) and the slope which is 4. We can plug it into:

Now simplify:

We can verify the result graphically: graph{ (y-x^2)(y-4x+4)=0 [-5, 5, -2, 12] }

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To find the derivative of f(x) = x^2 with respect to x, we can use the power rule. The power rule states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative is given by f'(x) = nx^(n-1).

Applying the power rule to f(x) = x^2, we have f'(x) = 2x^(2-1) = 2x.

To find the equation of the tangent line to y = x^2 at x = 2, we need to find the slope of the tangent line at that point. We can do this by evaluating the derivative at x = 2, which gives us f'(2) = 2(2) = 4.

The slope of the tangent line is 4. Now, we need to find the y-coordinate of the point of tangency. Plugging x = 2 into the original function, we have f(2) = 2^2 = 4.

So, the point of tangency is (2, 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.

Using (x1, y1) = (2, 4) and m = 4, the equation of the tangent line is y - 4 = 4(x - 2). Simplifying, we get y = 4x - 4.

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