Question

Derivatives of Exponential and Logarithmic Functions 179 Let \(f(x)=2^{x}\)\ (a) Find \(f^{\prime}(0) . \quad\) ln 2 (b) Use the definition of the derivative to write \(f^{\prime}(0)\) as a limit (c) Deduce the exact value of \(\lim _{h \rightarrow 0} \frac{2^{h}-1}{h}\) (d) What is the exact value of \(\lim _{h \rightarrow 0} \frac{7^{h}-1}{h} ?\)

Short answer

The values are (a) \(\ln 2\), (b) \(\lim_{h \rightarrow 0} \frac{2^{h}-1}{h}\), (c) \(\ln 2\), and (d) \(\ln 7\)

Step-by-step solution

Evaluate the derivative at a point

First, the derivative of \(f(x) = 2^x\) is \(f'(x) = 2^x \ln 2\). This is a commonly known derivative of the exponential function. To evaluate this derivative at \(x=0\), plug in \(0\) for \(x\) in the derivative function. \(f'(0) = 2^0 \ln 2 = \ln 2\)

Derive the limit function

The derivative of \(f(x)\) at \(0\) can also be written in terms of a limit as: \(f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{2^h - 2^0}{h} = \lim_{h \to 0} \frac{2^h - 1}{h}\)

evaluate the limit

The limit as \(h\) approaches \(0\) for \(\frac{2^h - 1}{h}\) function have been calculated in the previous steps, and it equals to \(\ln 2\)

Apply the limit property to a different base

Now, apply this same limit process to a different base, \(7^h\). \(\lim_{h \to 0} \frac{7^h - 1}{h} = \ln 7\) This is because the derivative of \(a^h\) is \(a^h \ln a\), so the limit as \(h\) approaches \(0\) is the derivative at \(0\), which is \(\ln a\).

Key concepts

Limit Definition of the Derivative

Understanding the derivative of a function is a fundamental concept in calculus, which describes the rate at which a function's output changes with respect to changes in its input. A more precise way to define the derivative of a function at a point is by using the limit definition. The derivative, denoted as f'(x), is defined as the limit of the average rate of change of the function over an interval as that interval's length approaches zero.
This can be mathematically expressed as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
In this definition, h represents a small change in x, and the ratio \(\frac{f(x+h) - f(x)}{h}\) represents the average rate of change of f over the interval from x to x + h. As h approaches zero, this ratio approaches the instantaneous rate of change, or the derivative, at the point x.

Exponential Function Derivative

An exponential function can be written in the form \(f(x) = a^x\), where a is a positive constant, and x represents the exponent. The derivative of an exponential function is unique because it results in a function proportional to the original function itself. The general derivative form of an exponential function is given by:
\[ f'(x) = a^x \ln(a) \]
Here, \ln(a) is the natural logarithm of a, representing the constant of proportionality. To derive this formula, we rely on the properties of natural logarithms and on the limit definition of the derivative. The derivative tells us how the function's value changes with respect to a change in x, and for exponential functions, this change is proportional to the function's current value, scaled by the natural logarithm of the base.

Natural Logarithm Properties

The natural logarithm, commonly represented as ln, is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm has several important properties that are frequently used in calculus, including:

• \(\ln(1) = 0\): The logarithm of one is zero.
• \(\ln(a \cdot b) = \ln(a) + \ln(b)\): The logarithm of a product is the sum of the logarithms.
• \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\): The logarithm of a quotient is the difference of the logarithms.
• \(\ln(a^r) = r \cdot \ln(a)\): The logarithm of a power is the exponent times the logarithm of the base.

These properties make it easier to work with expressions involving logarithms, particularly in finding derivatives of exponential functions.

L'Hôpital's Rule

In calculus, L'Hôpital's rule is a method for evaluating limits that result in indeterminate forms, such as \(0/0\) or \(\infty/\infty\). When faced with a limit that cannot be directly computed, L'Hôpital's rule suggests that we can find the limit of the ratio of the derivatives of the numerator and the denominator, under certain conditions. The rule states that if the limits of the functions f(x) and g(x) as x approaches c are both zero or both infinity, and the derivatives f'(x) and g'(x) are continuous near c, then:
\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \]
provided the limit on the right side exists or is infinity. This powerful rule simplifies the calculation of limits and is especially useful in more complex situations where the limit of a quotient is not immediately apparent.