Calculus of a Single Variable, Chapter 2, 2.4, Section 2.4, Problem 75

You need to find the equation of the tangent line to the given curve, at the point (-1,1), using the formula:
f(x) - f(-1) = f'(-1)( - (-1))
You need to put y = f(x) and you need to notice that f(-1) = 1.
You need to evaluate the derivative of the given function, using chain rule, such that:
f'(x) = ((4x^3 + 3)^2)' => f'(x) = 2(4x^3 + 3)*(4x^3 + 3)'
f'(x) = 2(4x^3 + 3)*(12x^2)
You need to evaluate f'(x) at x = -1, hence, you need to replace -1 for x in equation of derivative:
f'(-1) = 2(4(-1)^3 + 3)*(12(-1)^2)
f'(-1) = 24(-4 + 3) => f'(-1) = -24
You need to replace the values into equation of tangent line, such that:
f(x) - 1 = -24(x + 1) => y = -24x - 24 + 1 => y = -24x - 23
Hence, evaluating the equation of the tangent line to the given curve, at the given point, yields y = -24x - 23 .