Question

Use the properties of logarithms to simplify the following functions before computing $f^{\prime }(x)$. $$f(x)=\ln \frac{(2x-1)(x+2{)}^3}{(1-4x{)}^2}$$

Short answer

Question: Use the properties of logarithms to simplify the function $$f(x)=\ln \frac{(2x-1)(x+2{)}^3}{(1-4x{)}^2}$$ and then find its derivative, $$f^{\prime }(x)$$. Answer: The simplified function is $$f(x)=\ln (2x-1)+3\ln (x+2)-2\ln (1-4x)$$ and its derivative is $$f^{\prime }(x)=\frac{2}{2x-1}+\frac{3}{x+2}+\frac{8}{1-4x}$$.

Step-by-step solution

Apply the properties of logarithms to simplify the function

Using property 2 of logarithms, we can write the given function as: $$f(x)=\ln (2x-1)+3\ln (x+2)-2\ln (1-4x)$$

Compute the derivatives of simpler functions

We will now compute the derivatives of each of the three simpler logarithmic functions: 1. If $g(x)=\ln (2x-1)$, then $g^{\prime }(x)=\frac{1}{2x-1}\ast \frac{d}{dx}(2x-1)=\frac{2}{2x-1}$ 2. If $h(x)=\ln (x+2)$, then $h^{\prime }(x)=\frac{1}{x+2}\ast \frac{d}{dx}(x+2)=\frac{1}{x+2}$ 3. If $j(x)=\ln (1-4x)$, then $j^{\prime }(x)=\frac{1}{1-4x}\ast \frac{d}{dx}(1-4x)=-\frac{4}{1-4x}$

Compute the derivative of simplified function

Using the chain rule, we can compute the derivative of the simplified function as follows: $$f^{\prime }(x)=\frac{d}{dx}(\ln (2x-1)+3\ln (x+2)-2\ln (1-4x))$$ $$f^{\prime }(x)=\frac{2}{2x-1}+3\frac{1}{x+2}-2(-\frac{4}{1-4x})$$

Simplify the derivative

Combine terms with the same denominator and simplify: $$f^{\prime }(x)=\frac{2}{2x-1}+\frac{3}{x+2}+\frac{8}{1-4x}$$ This is the simplified derivative of the given function.

Key concepts

Properties of Logarithms

The properties of logarithms are powerful tools that allow us to simplify expressions involving logarithmic functions. Let's understand them one by one:

• The product property states that ${\log }_b(MN)={\log }_{b}M+{\log }_{b}N$, which indicates that the logarithm of a product is the sum of the logarithms.
• The quotient property is ${\log }_b\left(\frac{M}{N}\right)={\log }_{b}M-{\log }_{b}N$. Here, the logarithm of a quotient becomes the difference of the logarithms.
• The power property states ${\log }_b(M^p)=p\cdot {\log }_{b}M$, which can simplify the logarithm of a power as a multiple of that logarithm.

In our exercise, we applied these properties to transform \(f(x)=\ln \frac{(2x-1)(x+2)^3}{(1-4x)^2}\) into its simplified form. Using these properties helped decompose a complex logarithmic expression into several manageable ones.

Chain Rule

The chain rule is an essential method in calculus for finding the derivative of composite functions. It's like unwrapping layers of an onion to get to the center. Whenever you have a function that is composed of an outer function and an inner function, you'll likely need the chain rule. In its basic form, for a function \(y = f(g(x))\):

• The derivative, using the chain rule, is given by $\frac{dy}{dx}=f^{\prime }(g(x))\cdot g^{\prime }(x)$.

In our step-by-step solution, while computing the derivatives of simpler logarithmic functions, we utilized the chain rule:

• For instance, going from $\ln (2x-1)$ to $\frac{2}{2x-1}$, we used the chain rule to differentiate the inner function $2x-1$ before applying the logarithmic derivative.

This process ensured we accounted for the derivative of both the inner and outer functions.

Simplifying Complex Functions

Simplifying complex functions is crucial for accurate and manageable calculations. By reducing complexity, we can easily apply rules and find solutions. Take our initial function, \(f(x)=\ln \frac{(2x-1)(x+2)^3}{(1-4x)^2}\). Without simplification, directly computing derivatives can be cumbersome and error-prone.There are several steps to simplifying complex functions:

• Apply algebraic identities: Use the properties of algebra to rewrite expressions in simpler forms.
• Use properties of functions: In the case of logarithms, use sum, difference, and power rules to break down the expression.
• Factor and cancel terms where possible: Identify common factors you can cancel out or combine.

This process not only makes derivative calculations easier but also helps in visualizing and understanding the behavior of the function.

Logarithmic Differentiation

Logarithmic differentiation is a technique particularly useful for functions composed of products, quotients, or powers of variables. By taking the natural logarithm of both sides of an equation, differentiation becomes more straightforward.Why use logarithmic differentiation?

• It simplifies the differentiation of complex functions.
• By reducing multiplicative complexities to additive ones, we use simpler derivatives of logarithmic functions.

Let's see how it’s applied in our solution:

• First, take the logarithm: Start with $\ln (y)=\ln \left(\frac{(2x-1)(x+2{)}^3}{(1-4x{)}^2}\right)$.
• Differentiate implicitly with respect to x: This gives us simpler expressions to work with, leading to derivatives like $\frac{2}{2x-1}+\frac{3}{x+2}+\frac{8}{1-4x}$.

This simplification makes these complex operations much more manageable and accurate.