How do I answer each step of this calculus problem?

Question

Consider the cube root function 𝑓(𝑥) = 𝑥^1/3

Find the equation of the line tangent to this function at the point 𝑥 = 8.

Use the tangent line to find an approximate value to 9^1/3, the cube root of 9.

Answer 1

The derivative of the function y= x^(1/3) will give you the slope of the tangent line

Use the power rule to calculate the derivative which is (1/3)x^(-2/3) or (1/3)/x^(2/3

When x=8 this is (1/3)/(64^(1/3)= 1/12

When x = 8 y= 2

so 2=(1/12)(8) + b

solve for b. b= 4/3 so the equation of the tangent line is y= (1/12)x+ 4/3

graph this line and see what the value of y is when x=9 It is 2.08 which is a very close approximation of the cube root of 9