If y = (x^2 + 1)^5, find dy/dx at x = 1.

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Question: If y = (x^2 + 1)^5, find dy/dx at x = 1.

Options:

Correct Answer: 20

Solution:

dy/dx = 5(x^2 + 1)^4(2x). At x = 1, dy/dx = 5(2)^4(2) = 5(16)(2) = 160.

If y = (x^2 + 1)^5, find dy/dx at x = 1.

If y = (x^2 + 1)^5, find dy/dx at x = 1.

Correct Answer: 160

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)).
Step 7: Calculate (1)^2 + 1 = 2, so dy/dx = 5(2)^4 * 2.
Step 8: Calculate (2)^4 = 16, so dy/dx = 5 * 16 * 2.
Step 9: Finally, calculate 5 * 16 * 2 = 160.

Chain Rule – The problem tests the application of the chain rule in differentiation, particularly for composite functions.
Power Rule – The power rule is used to differentiate the outer function raised to a power.
Substitution – Substituting the value of x into the derivative to find the specific slope at that point.