dy/dx = 5(x^2 + 1)^4(2x). At x = 1, dy/dx = 5(2)^4(2) = 5(16)(2) = 160.
Questions & Step-by-step Solutions
1 item
Q
Q: If y = (x^2 + 1)^5, find dy/dx at x = 1.
Solution: dy/dx = 5(x^2 + 1)^4(2x). At x = 1, dy/dx = 5(2)^4(2) = 5(16)(2) = 160.
Steps: 9
Step 1: Identify the function y = (x^2 + 1)^5.
Step 2: Use the chain rule to find the derivative dy/dx. The chain rule states that if you have a function of a function, you multiply the derivative of the outer function by the derivative of the inner function.
Step 3: The outer function is (u)^5 where u = (x^2 + 1). The derivative of (u)^5 is 5u^4.
Step 4: The inner function u = (x^2 + 1). The derivative of u with respect to x is 2x.
Step 5: Combine the derivatives using the chain rule: dy/dx = 5(x^2 + 1)^4 * (2x).
Step 6: Substitute x = 1 into the derivative: dy/dx = 5((1)^2 + 1)^4 * (2(1)).
