Let f(x) = 4 cubed root(x). Compute f ' (64) and find an equation of the tangent line to the graph x=64. (Use symbolic notation and fractions where needed.

Question

Let y= f(x) and express the equation in terms of x and y.

f ' (64) = ?

Equation: ?

Answer 1

When x = 64, f(x) = 4. So you have a point (64,4) on the curve AND the tangent line - otherwise it would not be tangent at x=64. Take the derivative to find the slope. f'(x) = 1/(3x^(2/3)) by the power rule. When x 64, the slope (plug this into your derivative) is 1/48. So you now have a point and a slope.

y = mx+b

4 = 1/48 * 64 + b

4 = 4/3 + b

b = 8/3

y = x/48 + 8/3