Determine whether the following statements are true and give an explanation or counterexample.
d/dx((√2)^x) = x(√2)^{x - 1}

Verified step by step guidance

Step 1: Recognize the function given is \( f(x) = (\sqrt{2})^x \). This is an exponential function where the base is \( \sqrt{2} \).

Step 2: Recall the derivative rule for exponential functions of the form \( a^x \), which is \( \frac{d}{dx}(a^x) = a^x \ln(a) \).

Step 3: Apply the derivative rule to \( f(x) = (\sqrt{2})^x \). The derivative is \( (\sqrt{2})^x \ln(\sqrt{2}) \).

Step 4: Compare the derived expression \( (\sqrt{2})^x \ln(\sqrt{2}) \) with the given expression \( x(\sqrt{2})^{x-1} \).

Step 5: Conclude that the given statement is false because \( (\sqrt{2})^x \ln(\sqrt{2}) \neq x(\sqrt{2})^{x-1} \). The correct derivative involves the natural logarithm of the base, not a multiplication by \( x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of Exponential Functions

The derivative of an exponential function of the form a^x, where a is a constant, is given by d/dx(a^x) = a^x * ln(a). This rule is essential for differentiating functions where the base is a constant raised to a variable exponent, such as (√2)^x.

Power Rule

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is commonly used for differentiating polynomial functions and is not applicable to exponential functions where the base is a constant.

Counterexamples in Mathematics

A counterexample is a specific case that disproves a general statement. In this context, providing a counterexample to the statement d/dx((√2)^x) = x(√2)^(x - 1) would involve showing that the left-hand side does not equal the right-hand side for any value of x, thus demonstrating the statement's falsehood.

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