Question

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) for the following functions. $$f(x)=10 e^{x}$$

Short answer

Answer: The first, second, and third derivatives for the given function are: \(f'(x) = 10 e^x\), \(f''(x) = 10 e^x\), and \(f'''(x) = 10 e^x\).

Step-by-step solution

Find the first derivative \(f'(x)\)

To find the first derivative, apply the exponential function derivative rule which states that if \(f(x) = a e^{bx}\), then \(f'(x) = ab e^{bx}\). In this case, \(a = 10\) and \(b = 1\). Therefore, $$f'(x) = 10 \cdot 1 \cdot e^x = 10e^x$$

Find the second derivative \(f''(x)\)

Now, we need to find the second derivative, which involves differentiating the first derivative \(f'(x)\) that we have found in Step 1. Again, apply the exponential function derivative rule to \(f'(x) = 10 e^x\). In this case, \(a = 10\) and \(b = 1\). Therefore, $$f''(x) = 10 \cdot 1 \cdot e^x = 10e^x$$

Find the third derivative \(f'''(x)\)

Finally, find the third derivative by differentiating the second derivative \(f''(x)\) that we have found in Step 2. Once again, apply the exponential function derivative rule to \(f''(x) = 10 e^x\). In this case, \(a = 10\) and \(b = 1\). Therefore, $$f'''(x) = 10 \cdot 1 \cdot e^x = 10e^x$$ Now we have found the first, second, and third derivatives for the given function: $$f'(x) = 10 e^x, \quad f''(x) = 10 e^x, \quad f'''(x) = 10 e^x$$

Key concepts

The First Derivative

The first derivative is a foundational concept in calculus and is often used to understand various properties of a function. When we find the first derivative of a function, we are essentially calculating its rate of change or slope at any given point. For the function given, \(f(x) = 10e^x\), the first derivative is calculated using the rule for the derivative of an exponential function, \(f'(x) = ab e^{bx}\). Here, \(a = 10\) and \(b = 1\), which simplifies to:

• \(f'(x) = 10 \times 1 \times e^x = 10e^x\)

This result tells us that the slope or rate of change of the function at any point \(x\) is \(10e^x\). Breaking this down, the rate of change is proportional to the value of the function itself, showing how exponentials grow or decay in a proportional manner. This property is unique to exponential functions and is very useful in modeling real-world phenomena such as population growth or radioactive decay.

Understanding the Exponential Function

The exponential function \(e^x\) plays a crucial role in calculus and many real-world applications. The constant \(e\) (approximately 2.718) is the base of the natural logarithm and is often used in mathematics to model continuous growth or decay. An important property of the exponential function is that its derivative is proportional to the function itself.

• This means that \(\frac{d}{dx}(e^x) = e^x\)
• This property simplifies calculations, especially when dealing with continuous growth models.

In our given function, \(f(x) = 10e^x\), the coefficient \(10\) scales the exponential function, affecting the magnitude of growth or decay while maintaining the same relative rate of change. This proportionality makes the exponential function ideal for modeling situations like interest compounding, and it's a critical concept to grasp for advanced studies in calculus and differential equations.

Exploring Higher-Order Derivatives

Higher-order derivatives refer to taking the derivative of a function multiple times. In calculus, the first derivative provides information about the slope of the function, while higher-order derivatives (such as the second and third) reveal additional insights like concavity and changes in curvature. For example, the second derivative can help determine if a function is concave up or down, guiding us in understanding its shape.

• In the case of \(f(x) = 10e^x\), the second derivative is \(f''(x) = 10e^x\).
• Continuing to the third derivative also yields \(f'''(x) = 10e^x\).

This repetition occurs because the exponential function maintains its form throughout differentiation. Thus, higher-order derivatives for exponential functions where the exponent is linear (like \(x\) here) lead to the same formula. Recognizing this pattern simplifies problems involving exponential functions in calculus, assisting in predicting behavior and applying these functions to real-world scenarios.