91Ó°ÊÓ

An exponential function is any function where a constant base is raised to a variable exponent. In the given problem, the function is represented as \(2^x\). Here's why it's called exponential:

• Constant Base: The base number here is 2, and it remains constant.
• Variable Exponent: The exponent is the variable \(x\), allowing the function to grow rapidly as \(x\) increases.

To find the derivative of an exponential function like \(a^x\), where \(a\) is a constant, the rule is: \(\frac{d}{dx}a^x = a^x \ln a\). This means you not only multiply the original function by the natural logarithm of the base \(a\).
For our function, \(\frac{d}{dx}2^x = 2^x \ln 2\). This derivative tells us the rate of change of the exponential function at any given \(x\) value. Exponential functions increase or decrease very quickly, depending on whether \(a\) is more than 1 or between 0 and 1. Understanding how to differentiate these is crucial because they model phenomena like population growth and radioactive decay.

Differentiating terms that look like \(x^n\) is where the power rule comes into play. The power rule is a central tool in calculus used to find derivatives of polynomial functions, including those with negative exponents.
To apply the power rule, follow this formula: \(\frac{d}{dx}x^n = n x^{n-1}\). This involves multiplying the original exponent \(n\) by the coefficient in front of \(x^n\) and decreasing the exponent by one.
In our problem, we first rewrite the rational term \(\frac{2}{x^3}\) as \(2x^{-3}\) to apply the power rule more easily. The derivative then becomes \(-6x^{-4}\) because:

• The original exponent \(-3\) gets multiplied by the coefficient 2, resulting in \(-6\).
• The new exponent is \(-3 - 1 = -4\).

This simplified approach makes it easier to handle derivatives of terms presented as quotients or fractions. The power rule is an efficient method to differentiate any polynomial term.

A rational function is typically expressed as the quotient of two polynomials \(\frac{P(x)}{Q(x)}\). In the problem, this is shown by \(\frac{2}{x^3}\), where the numerator is 2 (a constant and itself a polynomial of degree 0) and the denominator is \(x^3\).
Rational functions can often be rewritten to make differentiation easier. Here, we convert \(\frac{2}{x^3}\) into \(2x^{-3}\), which behaves similarly to polynomial terms for which the power rule can be applied.

• Expressing it as \(2x^{-3}\) allows direct application of the power rule, which simplifies finding the derivative.
• The negative exponent also indicates the function decreases in value as \(x\) becomes larger.

Rational functions can be more complex when the numerator or the denominator becomes a higher degree polynomial, but the same principles apply. Understanding how to convert and manage these is vital to solving real-world scenarios involving rates of change and limits.