Question

Calculate the derivative of the following functions (i) using the fact that \(b^{x}=e^{x \ln b}\) and (ii) by using logarithmic differentiation. Verify that both answers are the same. $$y=3^{x}$$

Short answer

Question: Calculate the derivative of the function \(y=3^x\) using two different methods and verify that the answers are the same. Answer: The derivative of the function \(y=3^x\) is \(\frac{dy}{dx} = 3^x\ln3\), which was found using both methods (using the fact that \(3^x = e^{x\ln3}\) and using logarithmic differentiation).

Step-by-step solution

Method 1: Using \(3^x = e^{x\ln3}\)

To find the derivative of the function \(y=3^x\), we first rewrite \(3^x\) as \(e^{x\ln3}\). Then we can determine the derivative using the chain rule. The chain rule states that if we have a composition of functions, say \(u(x)=f(g(x))\), then the derivative is given by \(u'(x)=f'(g(x))g'(x)\). So, rewrite the given function: $$y=e^{x\ln3}.$$ Now, apply the chain rule, where \(f(x)=e^x\) and \(g(x)=x\ln3\): $$ \frac{dy}{dx} = e^{x\ln3} \cdot \frac{d(x\ln3)}{dx}. $$ Now, differentiate the function \(x\ln3\) with respect to \(x\): $$ \frac{d(x\ln3)}{dx} = \ln3. $$ Plug this value back into our derivative expression: $$ \frac{dy}{dx} = e^{x\ln3} \cdot \ln3. $$ Finally, rewrite \(e^{x\ln3}\) back into the original function \(3^x\): $$ \frac{dy}{dx} = 3^x \cdot \ln3. $$ The derivative of the function \(y=3^x\) using this method is \(\frac{dy}{dx} = 3^x\ln3\).

Method 2: Logarithmic differentiation

We can also find the derivative of the function \(y=3^x\) using logarithmic differentiation. First, take the natural logarithm (ln) of both sides of the equation: $$ \ln y = \ln(3^x). $$ Using the property of logarithms, we can rewrite it as: $$ \ln y = x\ln3. $$ Now, differentiate both sides of the equation with respect to \(x\): $$ \frac{1}{y}\frac{dy}{dx} = \ln3. $$ Now multiply both sides by \(y\) to isolate \(\frac{dy}{dx}\): $$ \frac{dy}{dx} = y\ln3. $$ Finally, replace \(y\) with the original function \(3^x\): $$ \frac{dy}{dx} = 3^x\ln3. $$ The derivative of the function \(y=3^x\) using logarithmic differentiation is also \(\frac{dy}{dx} = 3^x\ln3\).

Verification

Since both methods yielded the same derivative, we can conclude that the derivative of the function \(y=3^x\) is indeed \(\frac{dy}{dx} = 3^x\ln3\).

Key concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to simplify the process of finding the derivative of certain functions, particularly those where direct differentiation is complex due to variable exponents or products of functions. It is advantageous when dealing with exponential functions or functions with variables in both the base and the exponent.

Here's how it works: We start by taking the natural logarithm of both sides of an equation involving our function, which brings down exponents and turns products into sums. This transformation often simplifies the differentiation process. After applying the natural logarithm, we use the derivative of the natural logarithm function, which is \( \frac{d}{dx}\text{ln}(x) = \frac{1}{x} \), to differentiate both sides with respect to x. Then, we solve for \( \frac{dy}{dx} \) by isolating it and multiplying both sides by \( y \), which is then replaced by the original function. The derivative obtained using logarithmic differentiation is powerful because it often makes complex differentiation more manageable.

Chain Rule

The chain rule is a fundamental rule in calculus for finding the derivative of composite functions. The chain rule states that to differentiate a composite function \( f(g(x)) \) with respect to \( x \), we take the derivative of the outer function \( f \) at the inner function \( g(x) \) and multiply by the derivative of the inner function g with respect to \( x \).

The formula for the chain rule is: \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \). It is like saying, 'First find how fast the inner function is changing (\( g'(x) \) ), and then find out how fast the outer function is changing as a result (\( f'(g(x)) \) ).

It is particularly useful for functions where one function is nested inside another, such as \( e^{x\ln3} \) where we have the exponential function \( e^x \) with the inner function \( x\ln3 \) nested inside. The chain rule simplifies the process of finding the derivative for such composite functions, making them easier to handle.

Natural Logarithm

The natural logarithm, denoted as \( \text{ln} \), is the logarithm to the base \( e \), where \( e \) is Euler's number, approximately equal to 2.71828. This special logarithm is the inverse function of the exponential function, meaning \( \text{ln}(e^x) = x \) and conversely, \( e^{\text{ln}(x)} = x \).

This unique characteristic of the natural logarithm makes it particularly useful in calculus for transforming multiplication into addition, turning powers into products, and moving exponents to the front, which is crucial when applying both the product rule and logarithmic differentiation. The derivative of \( \text{ln}(x) \) is \( \frac{1}{x} \) for \( x>0 \) and this property proves invaluable when differentiating functions involving logarithms.

Exponential Function Differentiation

Differentiation of exponential functions involves finding the rate of change or slope of the function \( e^{x} \) or its equivalent forms. In general, the derivative of an exponential function \( \text{a}^{x} \) is found using the base \( e \) expression \( \text{a}^{x} = e^{x\ln{a}} \) and then applying the chain rule. The reason for this base conversion is that the derivative of \( e^{x} \) with respect to \( x \) is simply \( e^{x} \) itself, making calculus operations smoother.

For any positive constant \( a \) and variable \( x \) the derivative, using the chain rule, is \( \frac{d}{dx}a^{x} = a^{x} \ln(a) \) which simplifies the process of differentiation immensely. It is worth noting that the exponential function's derivative is unique in that it maintains the original function's form, a fundamental property that is frequently utilized in mathematics and the sciences.