Problem 38 - Calculus of a Single Variable

Question

Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=5 e^{x^{2}+5} $$

Step-by-Step Solution

Answer

The derivative of the function \(y=5e^{x^{2}+5}\) is \(y'=10x \cdot e^{x^{2}+5}\).

Step 1: Identify the Inner Function and the Outer Function

Here, the inner function is \(x^{2}+5\), the part inside the exponent and the outer function is the exponential function \(5e^{u}\), where \(u\) is the inner function.

Step 2: First, Differentiate the Outer Function

The derivative of the outer function \(5e^{u}\) is \(5e^{u}\). Replace \(u\) with the inner function \(x^{2}+5\). So, the derivative of the outer function is \(5e^{x^{2}+5}\).

Step 3: Differentiate the Inner Function

The derivative of the inner function \(x^{2}+5\) is \(2x\).

Step 4: Apply the Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. So, multiply the results from steps 2 and 3: \[y'=5e^{x^{2}+5} \cdot 2x\].

Step 5: Simplify

Multiply the constants in the equation to get the final derivative: \[y'=10x \cdot e^{x^{2}+5}\].

Key Concepts

Chain Rule for Derivatives

The chain rule is a fundamental technique in calculus used when differentiating composite functions. When a function is composed of two or more functions, we can't simply differentiate each component separately. Instead, we apply the chain rule to find the derivative of the entire composition.
In essence, the chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

• Identify the inner and outer functions. For example, if you have a function \(y = f(g(x))\), \(g(x)\) is the inner function and \(f(u)\) is the outer function, where \(u = g(x)\).
• Dfferentiate both functions separately.
• Multiply the derivative of the outer function by the derivative of the inner function.

In our exercise, we found the derivative of \(5e^{u}\) and multiplied it by the derivative of \(x^2 + 5\), leading us to the solution.

Exponential Functions

Exponential functions are a key type of function in calculus and are widely used in modeling real-world phenomena. An exponential function is written in the form \(f(x) = a \, e^{bx}\), where \(e\) is Euler's number (approximately 2.718).

• They have the unique property that their rate of change is proportional to their value, making them ideal for modeling growth and decay processes.
• When differentiating exponential functions, the general form allows us to use the fact that \(\frac{d}{dx}(e^{u}) = e^{u} \, u'\), where \(u'\) is the derivative of the exponent.

In the given exercise, the exponential function is \(5e^{x^2 + 5}\). We apply differentiation to find how this function changes as \(x\) changes, using the chain rule because the exponent itself is a function of \(x\).

Differentiation Techniques

Differentiation techniques are various methods used to find the derivative of functions efficiently. Among these techniques, the chain rule and the power rule are particularly vital for solving problems involving composite and basic polynomial functions.

• The power rule (\(\frac{d}{dx}(x^{n}) = nx^{n-1}\)) is often used for polynomial functions. This comes into play when finding derivatives like that of \(x^{2}\).
• The chain rule, as discussed earlier, helps manage complexities when differentiating nested functions.
• Combining rules: Often, two or more differentiation techniques need to be combined, such as applying the chain rule with the power rule.

In our exercise, these techniques were crucial. We differentiated the inner function \(x^{2}+5\) using the power rule and then applied the chain rule to integrate the exponential outer function. This mastery of differentiation ensures you can handle a wide variety of functions efficiently.