Problem 35 - Single Variable Calculus

Question

Use logarithmic differentiation to find the derivative of the function. $$ y=3^{x} $$

Step-by-Step Solution

Answer

The derivative of the function \(y = 3^x\) using logarithmic differentiation is \(\frac{dy}{dx} = (3^x)\ln(3)\).

Step 1: Take the natural logarithm of both sides of the equation.

To make it easier to differentiate, we will take the natural logarithm (ln) of both sides of the equation. $$ \ln(y) = \ln(3^x) $$

Step 2: Apply the power rule of logarithms.

According to the power rule of logarithms, we can move the exponent to the front, as a constant multiplier. Thus, we get: $$ \ln(y) = x\ln(3) $$

Step 3: Differentiate both sides of the equation with respect to x.

Next, we will differentiate both sides of the equation concerning x. The derivative of $$\ln(y)$$ concerning x is $$\frac{1}{y}\frac{dy}{dx}$$, and the derivative of $$x\ln(3)$$ is simply $$\ln(3)$$ since it is a constant: $$ \frac{1}{y}\frac{dy}{dx} = \ln(3) $$

Step 4: Solve for $$\frac{dy}{dx}$$ to find the derivative.

Now we will multiply both sides of the equation by y: $$ \frac{dy}{dx} = y\ln(3) $$

Step 5: Replace y with the original function.

Finally, we will substitute back our original function $$y=3^x$$ into the equation. This allows us to find the derivative of the function concerning x: $$ \frac{dy}{dx} = (3^x)\ln(3) $$ This is the derivative of the original function $$y=3^x$$ using logarithmic differentiation.

Key Concepts

Derivative of Exponential Functions

Understanding the derivative of exponential functions is essential in calculus. Exponential functions have the form f(x) = a^x, where a is a positive constant, and x is an exponent. The base a can be any positive real number, not just e (the natural base). The general derivative of an exponential function can be expressed using the original function itself, which is a unique and powerful property.

When you differentiate an exponential function, the result is proportional to the function itself. For example, if we have f(x) = e^x, the derivative f'(x) is simply e^x. However, for other bases like 3 or 10, we must multiply by the natural logarithm of the base; hence, f'(x) = a^x ln(a). This fact is leveraged in logarithmic differentiation, which can simplify complex differentiation problems involving exponential terms with variable exponents, as shown in the provided exercise.

Power Rule of Logarithms

The power rule of logarithms is a technique used to simplify expressions before differentiation. It states simply that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number:

log_b(m^n) = n log_b(m)

The beauty of this rule lies in its ability to convert multiplicative relationships into additive ones, allowing for easier manipulation of the terms. This is particularly useful when dealing with the differentiation of functions involving exponents, as it enables the exponent to be brought down in front of the logarithm, transforming a potentially difficult problem into a more manageable one. In our exercise, this rule was applied to take the variable exponent out in front, greatly simplifying the process of finding the derivative.

Differentiation Techniques

Differentiation techniques are an assortment of methods used to calculate the derivative of a function. Each technique is designed to tackle a specific type of function or equation. Among these methods are the product rule, quotient rule, chain rule, and logarithmic differentiation, to name a few.

Logarithmic differentiation, the method employed in the exercise, is particularly useful when the function is a power of x, especially when x itself is an exponent. The steps typically involve taking the natural logarithm of both sides of the equation, applying the power rule of logarithms, differentiating implicitly, and then solving for the derivative of the original function. It's a powerful technique that can simplify otherwise cumbersome problems. The ability to switch between these techniques as needed allows students and mathematicians to approach complex calculus problems with flexibility and precision.