If y = 2^x, find dy/dx at x = 1.

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

Options:

Correct Answer: 0.693

Solution:

dy/dx = 2^x * ln(2). At x = 1, dy/dx = 2^1 * ln(2) = 2 * 0.693 = 1.386.

If y = 2^x, find dy/dx at x = 1.

If y = 2^x, find dy/dx at x = 1.

Correct Answer: 1.386

Step 1: Identify the function given in the question, which is y = 2^x.
Step 2: To find dy/dx, we need to use the rule for differentiating exponential functions. The derivative of a^x is a^x * ln(a), where a is a constant.
Step 3: In our case, a = 2. So, we apply the rule: dy/dx = 2^x * ln(2).
Step 4: Now, we need to find dy/dx specifically at x = 1. Substitute x = 1 into the derivative we found: dy/dx = 2^1 * ln(2).
Step 5: Calculate 2^1, which is 2. Then, find ln(2), which is approximately 0.693.
Step 6: Multiply the results from Step 5: dy/dx = 2 * 0.693.
Step 7: Finally, calculate the multiplication: 2 * 0.693 = 1.386.

Differentiation of Exponential Functions – The question tests the ability to differentiate the function y = 2^x using the formula dy/dx = a^x * ln(a) for exponential functions.
Application of Derivative at a Point – The question requires evaluating the derivative at a specific point (x = 1), which tests understanding of both differentiation and substitution.