9–61. Evaluate and simplify y'.


y = 2^x²−x

Verified step by step guidance

1. Step 1: Identify the function y = 2^{x^2 - x}. This is an exponential function where the base is 2 and the exponent is a quadratic expression in x.
2. Step 2: To find y', the derivative of y with respect to x, use the chain rule. The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x).
3. Step 3: Recognize that the outer function is an exponential function with base 2, so the derivative of 2^u with respect to u is 2^u * ln(2). Here, u = x^2 - x.
4. Step 4: Differentiate the inner function u = x^2 - x with respect to x. The derivative of x^2 is 2x, and the derivative of -x is -1, so u' = 2x - 1.
5. Step 5: Apply the chain rule: y' = (2^{x^2 - x} * ln(2)) * (2x - 1). This is the derivative of the given function y with respect to x.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this case, we need to differentiate the function y = 2^(x² - x) to find y'.

Chain Rule

The Chain Rule is a key technique in differentiation used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. For the function y = 2^(x² - x), applying the Chain Rule is essential to correctly differentiate the exponent.

Exponential Functions

Exponential functions are functions of the form f(x) = a^g(x), where a is a constant and g(x) is a function of x. The derivative of an exponential function can be computed using the formula f'(x) = a^g(x) * g'(x) * ln(a). Understanding how to differentiate exponential functions is crucial for evaluating y' in this problem.