A Tangent Line. Find the tangent line, in slope—intercept form, for f(x) at x =1. Show the details of the power rule when calculating the derivative.

Question

F(x) = -4x^4 - 5x^3 - 2x^2 + 8/sqrx + 6

Answer 1

The Power Rule is that to find the derivative, if f(x) = x^r, f'(x) = rx^(r-1). So F'(x) = -16x^3 - 15x^2 - 4x - 4/(x^3/2). 8/x^.5 = 8x^-.5, so its derivative is -4x^-1.5.

F(1) = -4 - 5 - 2 + 8 + 6 = 3.

F'(1) = -16 - 15 - 4 - 4 = -39, so the slope is -39 and f(1) = -39(1) + C = 3. So C = 3 + 39 = 42, and the tangent line is f(x) = -39x + 42.