Problem 32 - Calculus and its applications

Question

Differentiate. $$ f(x)=x^{7} e^{4 x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = e^{4x}(7x^6 + 4x^7) \).

Step 1: Identify the Rule

The function given is a product of two functions: a polynomial and an exponential function. We will use the product rule for differentiation. The product rule states: if \( u(x) \) and \( v(x) \) are differentiable functions, then \( (uv)' = u'v + uv' \). Let \( u(x) = x^7 \) and \( v(x) = e^{4x} \).

Step 2: Differentiate \( u(x) = x^7 \)

Differentiate \( u(x) = x^7 \) using the power rule for differentiation: \( \frac{d}{dx}[x^n] = n x^{n-1} \). So \( u'(x) = 7x^{6} \).

Step 3: Differentiate \( v(x) = e^{4x} \)

The derivative of an exponential function \( e^{ax} \) is \( ae^{ax} \). Therefore, \( v'(x) = 4e^{4x} \).

Step 4: Apply the Product Rule

Substitute \( u(x), u'(x), v(x), \) and \( v'(x) \) into the product rule formula: \[ f'(x) = u'(x)v(x) + u(x)v'(x) = 7x^{6}e^{4x} + x^{7}(4e^{4x}) \].

Step 5: Simplify the Expression

Combine like terms in the expression: \[ f'(x) = 7x^{6}e^{4x} + 4x^7e^{4x} \]. Factor out the common term \( e^{4x} \): \[ f'(x) = e^{4x}(7x^{6} + 4x^{7}) \].

Key Concepts

Product Rule

The product rule is a fundamental concept in calculus used to find the derivative of a product of two functions. In simple terms, it allows us to differentiate functions that are multiplied together. Here's how it works in a nutshell:

• Identify the two functions that are being multiplied. In our example, these are \( u(x) = x^7 \) and \( v(x) = e^{4x} \).
• The product rule states that the derivative of the product \( u(x)v(x) \) is \( (uv)' = u'v + uv' \).
• To apply the rule, differentiate each function separately (find \( u'(x) \) and \( v'(x) \)), then plug them into the formula.

By understanding the product rule, solving complex derivatives becomes easier and systematic. It is a powerful tool that extends the range of functions that we can differentiate.

Power Rule

The power rule is one of the simplest and most commonly used rules in differentiation. This rule helps to find the derivative of polynomial functions swiftly.Here's how it works:

• Given a function in the form \( x^n \), the power rule states that its derivative is \( nx^{n-1} \).
• In our exercise, we differentiate \( u(x) = x^7 \). Applying the power rule gives us: \( u'(x) = 7x^{6} \).
• This method can be applied to any polynomial term, making it a quick way to handle derivatives of basic power functions.

The power rule simplifies differentiation, allowing you to move swiftly through polynomial functions like in this example. Once mastered, it becomes second nature in calculus.

Exponential Function Derivative

The derivative of an exponential function is unique because it is proportional to the original function. Here’s a simple way to understand it:

• The derivative of an exponential function \( e^{ax} \) is \( ae^{ax} \), where 'a' is a constant.
• In our example, \( v(x) = e^{4x} \), the derivative is \( v'(x) = 4e^{4x} \).
• This rule highlights a key property of exponential functions: the rate of change of \( e^{ax} \) is always a constant multiple of itself.

Understanding how exponential functions behave under differentiation helps in grasping more complex functions involving exponential terms. This property allows exponential functions to be differentiated smoothly and quickly.