Let f(x)=x^3-x-1
a. find the equation of the tangent line to the graph of the function at x = 0.
b. what does the derivative of the function at x = 0 tell you about the direction at that point? is it increasing, decreasing, or neither?

To find the gradient of the tangent, we must first differentiate the function.



The gradient at x = 0 is given by evaluating f'(0).



The derivative of the function at this point is negative, which tells us the function is decreasing at that point.

The tangent to the line is a straight line, so we will have a linear equation of the form y = mx + c. We know the gradient, m, is equal to -1, so



Now we need to substitute a point on the tangent into this equation to find c. We know a point when x = 0 lies on here. To find the y-coordinate of this point we need to evaluate f(0).



So the point (0, -1) lies on the tangent. Substituting into the tangent equation: