Problem 36 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=e^{-2 x^{3}} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=e^{-2 x^{3}}\) is \((-6x^{2})e^{-2 x^{3}}\).

Step 1: Recognise the function type

The function is of the exponential form \(y=e^{f(x)}\) where \(f(x)=-2x^{3}\).

Step 2: Find the derivative of \(f(x)\)

The derivative of \(f(x)=-2x^{3}\) is \(f'(x) = -6x^{2}\), following the power rule of differentiation which states that the derivative of \(x^n\) is \(nx^{n-1}\).

Step 3: Apply the rule for the derivative of the exponential function

Now, applying the rule for the derivative of an exponential function \(y=e^{f(x)}\) which says that the derivative is \(f'(x)e^{f(x)}\), the derivative of \(y=e^{-2 x^{3}}\) becomes \((-6x^{2})e^{-2 x^{3}}\)

Key Concepts

Exponential Functions

Exponential functions are a special type of mathematical expressions where a number (often 'e', the base of the natural logarithms) is raised to a power. This power itself is another function of a variable, like \(x\). In this exercise, the function is \(y = e^{-2x^3}\). The key characteristic of exponential functions is the variable exponent, which can result in rapid growth or decay depending on the sign and value of the exponent.

• Base 'e': The number 'e' (approximately 2.718) is a mathematical constant which is the base for natural logarithms and is commonly used in exponential functions.
• Exponential Growth and Decay: When the exponent is positive, the function grows exponentially. If the exponent is negative, as in this exercise, it results in exponential decay.

Understanding how these functions behave is crucial when learning to differentiate them, as the nature of the change depends entirely on the exponent's function.

Power Rule

The power rule is a basic differentiation technique used to find the derivative of terms where the variable is raised to a constant power. When differentiating a function of the form \(x^n\), the power rule states that the derivative is \(nx^{n-1}\).

In this exercise, you encounter the function \(f(x) = -2x^3\) inside the exponential expression. Using the power rule, you find its derivative:

• Step 1: Take the exponent, \(n = 3\), and multiply it by the coefficient, giving \(-6\).
• Step 2: Subtract one from the exponent to obtain the new power, \(x^2\).
• Result: The derivative of \(-2x^3\) is \(-6x^2\).

This step is critical for applying further differentiation techniques, especially when dealing with compound functions.

Differentiation Techniques

Differentiation is the process of finding the rate at which a function is changing at any given point. It is a fundamental tool in calculus and vital for understanding behavior in mathematical functions. In the context of this exercise, a specific differentiation technique is applied to an exponential function.

To differentiate the function \(y = e^{f(x)}\), where \(f(x)\) is a function of \(x\), the derivative is calculated by:

• Finding the derivative of \(f(x)\), indicated by \(f'(x)\).
• Multiplying \(f'(x)\) by the original exponential function \(e^{f(x)}\).

This technique utilizes prior knowledge of the power rule. In the given problem,\(-6x^2\) (as derived from the power rule) is multiplied by the original expression \(e^{-2x^3}\) resulting in the derivative \(-6x^2 \cdot e^{-2x^3}\).

Mastery of these techniques not only simplifies complex functions but also lays the foundation for solving real-world problems involving rate of change and predicting outcome trends.