Problem 71 - Applied Calculus

Question

Find the derivative of each function. \(e^{x^{2}+5 x}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

We need to find the derivative of the given function: \(f(x) = e^{x^2 + 5x}\). This is an exponential function.

Step 2: Use the Chain Rule

The derivative of an exponential function \(e^{u}\) is \(e^{u} \cdot \frac{du}{dx}\). Let \(u = x^2 + 5x\).

Step 3: Differentiate the Inner Function

Find the derivative of \(u\) with respect to \(x\). For \(u = x^2 + 5x\), the derivative \(\frac{du}{dx} = 2x + 5\).

Step 4: Differentiate the Exponential Function

Apply the chain rule. The derivative of \(f(x) = e^{x^2 + 5x}\) is \(f'(x) = e^{x^2 + 5x} \cdot (2x + 5)\).

Step 5: Simplify the Expression

Combine the terms. The derivative of the function is \(f'(x) = (2x + 5) e^{x^2 + 5x}\).

Key Concepts

Chain Rule

The Chain Rule is an essential technique in calculus used to differentiate composite functions, which are functions made up of an outer and inner function. In the example of differentiating the exponential function \( f(x) = e^{x^2 + 5x} \), the Chain Rule helps us separate and tackle different parts.Here’s how it works:

• The outer function is \( e^u \) and the inner function is \( u = x^2 + 5x \).
• You first differentiate the outer function, treating the inner function as a constant, which results in \( e^u \).
• Then, you multiply this result by the derivative of the inner function, \( \frac{du}{dx} \), which we find to be \( 2x + 5 \).

Combining these steps gives us the derivative of the original function as \( f'(x) = e^{x^2 + 5x} \cdot (2x + 5) \). The Chain Rule lets us differentiate without directly handling the composite function as one piece.

Exponential Function

An exponential function is a type of mathematical function in the form \( f(x) = a^{g(x)} \), where the base \( a \) is a constant and the exponent is a function of \( x \). A common base is Euler's number, \( e \), which is approximately 2.718.When differentiating exponential functions, as in our example \( f(x) = e^{x^2 + 5x} \), the actual process involves:

• Maintaining the original exponential form \( e^{u} \).
• Multiplying by the derivative of the exponent \( u = x^2 + 5x \), which is calculated to be \( 2x + 5 \).

This method uses both the properties of exponential functions and the Chain Rule to find derivatives efficiently. This is why the original function's derivative is simply expressed with both components multiplied together.

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function changes. In simpler terms, it helps us compute the derivative of a function, which is essentially its slope or steepness at any given point.When we differentiate a function, such as \( u = x^2 + 5x \) in our example, the steps involve:

• Identifying the basic operations – addition, subtraction, multiplication, etc.
• Applying rules like the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \).
• Calculating \( 2x + 5 \) as the derivative of \( x^2 + 5x \) directly shows the effect of differentiation.

Differentiation is a powerful tool that, combined with the Chain Rule and understanding of exponential functions, allows for solving complex calculus problems efficiently.