`f(x) = 2/(root(4)(x^3)), (1,2)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
Eliminate the radical by rewriting it as a fraction.
The function becomes:
`f(x) = 2/ (x^3)^(1/4) = 2/x^(3/4) = 2x^(-3/4)`
Take the derivative by using the power rule.
`f'(x) = -3/4 (2)(x^(-3/4-1))`
`f'(x) = -3/2 (x^(-7/4))`
`f'(x) = -3/(2x^(7/4))`
`f'(1) = -3/(2(1)^(7/4)) = -3/2`
With the slope of the point, and the given point (1,2), use the slope intercept form to find the equation.
The equation of the tangent line is:
`y= -3/2 x +7/2`
Graph both the equation of the tangent line with the original function. They should intersect at (1,2).
See the image attached.