Calculus help

Question

Determine the equation of the line in slope-intercept form which is tangent to the following curve at x = 1.

 

f(x)=2/(³√x^2)

Answer 1

Write f(x) with a rational exponent.

f(x) = 2x^-2/3

Apply the power rule to determine f'(x)

f'(x) = 2(-2/3)x^-5/3 = (-4/3)x^-5/3

Determine f'(1) to find the slope of the tangent line at the point where x = 1.

f'(1) = (-4/3)(1)^-5/3 = -4/3

Plug x = 1 into the function definition to determine the y-coordinate of the point of tangency:

f(1) = 2 (1)^-2/3 = 2

The point of tangency is (1,2) and the slope of the tangent line at that point is -4/3.

y - 2 = (-4/3)(x - 1)

y = (-4/3)x + 4/3 + 6/3

y = (-4/3)x + 10/3

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