True or false: According to the power rule,
\[
\frac{d}{d x}\left[2^{x}\right]=x \cdot 2^{x-1}
\]

True

False

Answer

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Answer

Final Answer: \(\boxed{\text{False}}\)

Steps

Step 1 ：The power rule in calculus is a basic theorem that states that if we have a function of the form \(f(x) = x^n\), then the derivative of \(f(x)\) with respect to \(x\) is \(f'(x) = n*x^{(n-1)}\).

Step 2 ：However, in this case, the base is a constant (2) and the exponent is the variable (x), which is a different situation.

Step 3 ：The derivative of \(a^x\), where \(a\) is a constant, is actually \(a^x * ln(a)\).

Step 4 ：So, the derivative of \(2^x\) should be \(2^x * ln(2)\), not \(x * 2^{(x-1)}\).

Step 5 ：Therefore, the statement is false.

Step 6 ：Final Answer: \(\boxed{\text{False}}\)

True or false: According to the power rule,\[\frac{d}{d x}\left[2^{x}\right]=x \cdot 2^{x-1}\] True False

Answer

Popular Problems

True or false: According to the power rule,
\[
\frac{d}{d x}\left[2^{x}\right]=x \cdot 2^{x-1}
\]

True

False