Problem 21 - Calculus Single Variable

Question

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(2^{x}\)

Step-by-Step Solution

Answer

First derivative is \(2^x \ln(2)\), second is \(2^x (\ln(2))^2\), and third is \(2^x (\ln(2))^3\).

Step 1: Identify the Initial Function

The given function is \( f(x) = 2^x \). This is an exponential function with base 2.

Step 2: Compute the First Derivative

The first derivative of \( f(x) = 2^x \) with respect to \( x \) can be computed using the formula for the derivative of an exponential function: \( \frac{d}{dx} a^x = a^x \ln(a) \). Therefore, \( f'(x) = 2^x \ln(2) \).

Step 3: Compute the Second Derivative

To find the second derivative, differentiate the first derivative \( f'(x) = 2^x \ln(2) \). Applying the same derivative formula for exponential functions, we get \( f''(x) = 2^x (\ln(2))^2 \).

Step 4: Compute the Third Derivative

To find the third derivative, differentiate the second derivative \( f''(x) = 2^x (\ln(2))^2 \). Again, using the formula for derivatives of exponential functions, we have \( f'''(x) = 2^x (\ln(2))^3 \).

Key Concepts

Understanding the First Derivative

To grasp how to calculate the first derivative of an exponential function like twice raised to the power of x, we use a specific rule for differentiation. For a general exponential function of the form \( f(x) = a^x \), the first derivative is computed using the formula \( \frac{d}{dx} a^x = a^x \ln(a) \). Here, \( \ln \) represents the natural logarithm.

So, in the case of our function \( f(x) = 2^x \), the first derivative is simply \( f'(x) = 2^x \ln(2) \). This means that the rate at which our original function changes concerning x is \( 2^x \) multiplied by the natural logarithm of 2.

Exponential functions grow rapidly, which is why their derivatives significantly differ as the value of x changes. This first derivative tells us how steep the graph of the original function is at any point x.

Exploring the Second Derivative

Once we have determined the first derivative of the function, the next step is identifying the second derivative by differentiating the first derivative.

The process is similar: apply the exponential function differentiation rule again on \( f'(x) = 2^x \ln(2) \). We consider \( \ln(2) \) as a constant since it's not dependent on x, which allows us to apply the rule directly to the exponential part. The second derivative, therefore, becomes \( f''(x) = 2^x (\ln(2))^2 \).

This derivative illustrates how the rate of change itself is changing. In physical terms, if the first derivative represents velocity, then the second derivative would represent acceleration.

Whenever you compute a second derivative, it provides information over the concavity or convexity of the original function. In this instance, our exponential function \( 2^x \) continues its steep growth, as indicated by the consistent presence of \( 2^x \) in all calculated derivatives.

Analyzing the Third Derivative

Diving even deeper, the third derivative requires us to differentiate the second derivative.

Here we differentiate \( f''(x) = 2^x (\ln(2))^2 \) by considering \( (\ln(2))^2 \) as a constant and applying the exponential differentiation rule one more time. Consequently, the third derivative is calculated as \( f'''(x) = 2^x (\ln(2))^3 \).

The third derivative can be insightful in various contexts, such as understanding the "jerk" or change in acceleration in physics scenarios. For functions, it sometimes helps identify inflection points or measures of how the function's rate of change is itself changing at a second level.

In summary, utilizing derivatives helps us explore how any function behaves. By applying basic derivative rules pertinent to exponential functions, like our \( 2^x \), we can construct a detailed understanding of how the function evolves as x increases.