If (x) =3x squared - x cubed, find f prime(1) and use it to find an equation of the tangent line to the curve y=3(x) squared - x cubed at a point (1,2)
To find f'(1), we need to take the derivative of the function f(x) = 3x^2 - x^3 with respect to x.
Step 1: Take the derivative of each term separately:
The derivative of 3x^2 is 6x.
The derivative of -x^3 is -3x^2.
Step 2: Combine the derivative terms:
f'(x) = 6x - 3x^2.
Now let's find f'(1) by substituting x = 1 into the derivative function:
f'(1) = 6(1) - 3(1^2) = 6 - 3 = 3.
Now that we have f'(1) = 3, we can find the equation of the tangent line to the curve y = 3x^2 - x^3 at the point (1,2).
The equation of a tangent line can be given by the point-slope formula:
y - y1 = m(x - x1)
where (x1, y1) is the point on the curve and m is the slope of the tangent line.
Using the point (1,2) and the slope f'(1) = 3, we can substitute these values into the formula:
y - 2 = 3(x - 1)
Simplifying the equation:
y - 2 = 3x - 3
y = 3x - 1
Therefore, the equation of the tangent line to the curve y = 3x^2 - x^3 at the point (1,2) is y = 3x - 1.