Question

If f(u) = u⁴ - u - 5 and u = g(x) = 2/x, find the value of (f ∘ g)'(x) at x = -2.

Answer 1

You find the derivative (f \circ g)'(x) by first creating the composite function f(g(x)) and then differentiating and simplifying that expression. Finally, you evaluate the derivative at x = -2 to get the desired value.

Step-by-step explanation:

To find the value of (f \circ g)'(x) at x = -2, we need to first compute the composite function (f \circ g)(x) and then differentiate it with respect to x. The composite function is obtained by substituting g(x) into f(u), which yields f(g(x)) = f(2/x) = (2/x)^4 - (2/x) - 5. The derivative of the composite function, (f \circ g)'(x), is then found using the chain rule.

Here are the steps to find (f \circ g)'(x) and its value at x = -2:

1. Compute the composite function: f(g(x)) = (2/x)^4 - (2/x) - 5.
2. Take the derivative with respect to x using the chain rule: (f \circ g)'(x) = d/dx[(2/x)^4] - d/dx[2/x] - d/dx[5].
3. Simplify the expression for (f \circ g)'(x) using the power rule and the quotient rule.
4. Evaluate (f \circ g)'(x) at x = -2.