Step-by-step explanation:
Recall that the first derivative of an equation gives us the SLOPE for the tangent line.
So first, we should find the first derivative.
[tex]\frac{dy}{dx} = 6 - 3x^2[/tex]
With the first derivative, we can solve for the SLOPE of the tangent line at (1,5).
Using x = 1.
[tex]slope = 6 - 3(1)^2[/tex]
A tangent line is a line. And has the form [tex]y = mx +b[/tex]
Where m is the SLOPE of the line. and b is the y-intercept.
To get to this equation, let's use the point-slope technique.
[tex]y-y_1 = m(x-x_1)[/tex]
[tex]y - 5 = m(x-1)[/tex]
Solve for slope, substitute it into this equation for m and solve. That is the equation of the tangent line at the point (1,5).
