Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=e^{5 x}(1+2 x)^{6} $$

Step-by-Step Solution

Answer

The derivative of \(f(x)=e^{5 x}(1+2 x)^{6}\) is \(f^{\prime}(x) = 5e^{5x}(1+2x)^6 + e^{5x} \cdot 12(1+2x)^5\).

Step 1: Identify the functions

Firstly, recognize the given function as a product of two functions \(f(x) = f_{1}(x) \cdot f_{2}(x)\), where \(f_{1}(x) = e^{5x}\) and \(f_{2}(x)=(1+2x)^6\). The derivative of this function will be found using the product rule.

Step 2: Derive \(f_{1}(x) = e^{5x}\)

Applying the chain rule, the derivative of \(e^{5x}\) will be \(f_{1}^{\prime}(x) = 5 e^{5x}\). The derivative was found by multiplying e to the power of 5x by the derivative of 5x, which is 5.

Step 3: Derive \(f_{2}(x) = (1+2x)^6\)

According to the chain rule, the derivative of \((1+2x)^6\) will be \(f_{2}^{\prime}(x) = 6(1+2x)^5 \cdot 2\), which simplifies to \(f_{2}^{\prime}(x) = 12(1+2x)^5\). This derivative was found by multiplying the function by the derivative of 1+2x, which is 2, and simultaneously reducing the power by 1.

Step 4: Apply the Product Rule

According to the product rule, the derivative of the original function \(f(x) = e^{5x}(1+2x)^6\) can be obtained by using the formula \(h^{\prime}(x) = f_{1}^{\prime}(x) \cdot f_{2}(x) + f_{1}(x) \cdot f_{2}^{\prime}(x)\). Therefore, \(f^{\prime}(x) = 5e^{5x}(1+2x)^6 + e^{5x} \cdot 12(1+2x)^5\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus that allows us to differentiate composite functions. Composite functions consist of one function nested inside another. When differentiating such functions, the chain rule provides a systematic way to handle the derivative.To use the chain rule, we differentiate the outer function while keeping the inner function unchanged, then multiply this by the derivative of the inner function. This rule is typically written as:

• If we have a function in the form of \(g(h(x))\), the derivative is given by \((g(h(x)))' = g'(h(x)) \cdot h'(x)\) .

In our exercise, we applied the chain rule to differentiate \(e^{5x}\), where we treated \(5x\) as the inner function. The derivative is \(5 e^{5x}\) because we differentiate \(e^{5x}\) with respect to \(5x\) and multiply by the derivative of \(5x\), which is 5.Similarly, for \((1+2x)^6\), the outer function is raised to the power of 6, and the inner function is \(1 + 2x\). Differentiating gives \(6(1+2x)^5 \cdot 2\), since we take the power rule on the outside and multiply by the derivative of the inside.

Product Rule

The Product Rule is an essential tool for finding the derivative of functions that are products of two simpler functions. When we have a function \(f(x) = u(x) \cdot v(x)\), the product rule enables the calculation of its derivative.The formula for the product rule is:

• \(h'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)\)

This involves differentiating one function while keeping the other constant, then switching roles and summing the results.In our solved exercise, we identified \(e^{5x}\) as \(f_1(x)\) and \( (1+2x)^6\) as \(f_2(x)\). To find \(f'(x)\), we applied the product rule: firstly, multiply the derivative of \(f_1(x)\) with \(f_2(x)\) as is, and secondly, multiply \(f_1(x)\) as is, with the derivative of \(f_2(x)\). The results were then summed up to give the total derivative of the function.

Exponential Functions

Exponential functions form a vital class of functions in mathematics, defined by the expression \(e^{x}\), where \(e\) is Euler's number approximately equal to 2.71828. These functions exhibit rapid growth and are ubiquitous in various scientific calculations.A key feature of exponential functions is that their derivatives are proportional to the original function itself. For the function \(e^{5x}\), its derivative is \(5e^{5x}\), combining the original exponential function with the chain rule.Exponential functions are particularly useful due to their simplicity in differentiation. The aforementioned feature remains true even when the exponent itself is a complicated expression, as long as we apply the chain rule correctly to account for the inner function.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers, and they form the backbone of algebraic operations. A typical polynomial function looks like \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0\). Differentiating polynomials entails straightforward application of the power rule: \(d/dx[x^n] = n x^{n-1}\).In our exercise, \( (1+2x)^6\) acts like a polynomial function due to the integer power of 6. However, we employed the chain rule since it is a composite function involving a polynomial inside a power. After the chain rule, the derivative came out to be \(12(1+2x)^5\). This highlights the key characteristic of polynomial differentiation, where powers decrease by one, and coefficients get multiplied by the original power.Understanding how to differentiate polynomial functions efficiently allows deeper insight into calculus and aids in tackling more complex expressions.