limx →0 ((2^m+x)^1 / m-(2^n+x)^1 / n)/(x) is equal to (A) (1)/(m 2^m)-(1)/(n 2^n) (B) (1)/(m 2^m

Step 1: We are given the limit $lim _{x rightarrow 0} frac{left(2^{m}+xright)^{1 / m}-left(2^{n}

Limx →0 ((2^m+x)^1 / m-(2^n+x)^1 / n)/(x) is equal to (A)...

$\lim _{x \rightarrow 0} \frac{\left(2^{m}+x\right)^{1 / m}-\left(2^{n}+x\right)^{1 / n}}{x}$ is equal to
(A) $\frac{1}{m 2^{m}}-\frac{1}{n 2^{n}}$
(B) $\frac{1}{m 2^{m}}+\frac{1}{n 2^{n}}$
(C) $\frac{1}{m 2^{m-1}}-\frac{1}{n 2^{n-1}}$
(D) None of these

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Key Concepts

Limit Definition of the Derivative

This concept interprets a limit of the difference quotient as the derivative of a function at a particular point. It provides the foundation for finding the instantaneous rate of change by considering the behavior of the function as the change in the input approaches zero.

Power Rule

The power rule is a fundamental principle in differential calculus used to differentiate functions of the form x^p. It states that the derivative is given by multiplying the exponent by the original function raised to one power less, facilitating quick and straightforward differentiation of polynomial power functions.

Chain Rule

The chain rule is an essential technique for differentiating composite functions, where one function is nested within another. It involves differentiating the outer function evaluated at the inner function and then multiplying by the derivative of the inner function, enabling the differentiation of complex expressions.

Algebraic Simplification in Limit Computations

Algebraic simplification involves manipulating expressions—such as expanding, factoring, or canceling terms—to make limits and derivative computations more manageable. This step is crucial in resolving indeterminate forms and arriving at a simplified expression that can be directly evaluated.

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