Find f'(x) if f(x)=2x²-3x. Use this result to help you find the equation of the tangent line that passes through (3,f(3)). Give your equation in slope intercept form.

Answer 1

Hi, here is how we can proceed.

The derivative:

f(x) = 2x² - 3x

f'(x) = 2 • 2x - 3.

To calculate the derivative, we used the power rule, which says that the power function derivative is the power multiplied by the variable raised to power minus 1. We have two powers; I am showing one of them here (x²)' = 2x.

Next, we need to calculate f(3).

f(3) =2(3)² - 3(3)

= 18 - 9 = 9.

So, the point is (3, 9).

Here is the general equation in point-slope formula:

y - y_o = m(x - x_o).

In our case, x_o=3, y_o=9. The value of m is the value of the derivative at that point:

f'(3) = 4(3) - 3 = 9.

The equation in point-slope form is

y - 9 = 9 (x - 3).

To find the intercept-slope formula, we solve for y (steps omitted, can work out for practice):

y = 9x - 18.