If y = (x^2 + 1)^5, find dy/dx.
Correct Answer: dy/dx = 10x(x^2 + 1)^4
- Step 1: Identify the function y = (x^2 + 1)^5.
- Step 2: Recognize that this is a composite function, where the outer function is u^5 and the inner function is u = (x^2 + 1).
- Step 3: Apply the chain rule, which states that dy/dx = dy/du * du/dx.
- Step 4: Calculate dy/du. Since y = u^5, the derivative dy/du = 5u^4.
- Step 5: Substitute u back in: dy/du = 5(x^2 + 1)^4.
- Step 6: Now calculate du/dx. Since u = (x^2 + 1), the derivative du/dx = 2x.
- Step 7: Combine the results from Step 5 and Step 6 using the chain rule: dy/dx = dy/du * du/dx = 5(x^2 + 1)^4 * 2x.
- Step 8: Simplify the expression: dy/dx = 10x(x^2 + 1)^4.
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