Problem 24 - Applied Calculus

Question

Differentiate the functions in Problems 1-28. Assume that \(A\), \(B\), and \(C\) are constants. \(D=10-\ln p\)

Step-by-Step Solution

Answer

\( \frac{dD}{dp} = -\frac{1}{p} \)

Step 1: Identify the Function

The given function is \( D = 10 - \ln p \). We are required to find the derivative of this function with respect to \( p \).

Step 2: Differentiate Constant Term

Derivative of a constant term with respect to any variable is zero. So, the derivative of 10 with respect to \( p \) is 0.

Step 3: Differentiate the Logarithmic Term

The derivative of \( \ln p \) with respect to \( p \) is \( \frac{1}{p} \).

Step 4: Apply Derivative Rules

The derivative of \( D = 10 - \ln p \) is the derivative of each term separately. Combine the derivatives: \( \frac{dD}{dp} = 0 - \frac{1}{p} \).

Step 5: Simplify the Result

Simplify the expression from the previous step to get the final derivative: \( \frac{dD}{dp} = -\frac{1}{p} \).

Key Concepts

Derivative of logarithmic functions

When dealing with logarithmic functions, especially natural logarithms, understanding their derivative can unlock many calculus problems. The natural logarithm, denoted as \(\ln(x)\), has a simple derivative. It's crucial to remember,

• The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\).
• This rule applies as long as \(x\) is positive because the logarithm is only defined for positive values of \(x\).

For example, if you encounter a function like \(\ln(p)\), its derivative concerning \(p\) is simply \(\frac{1}{p}\). This straightforward rule helps in solving various calculus problems involving logarithmic differentiation.

Derivative rules

Derivative rules are handy tools that streamline the process of finding derivatives of complex problems. Some of the fundamental rules include:

• Constant Rule: The derivative of a constant term is always zero. This simplifies calculations right from the start. For instance, the derivative of a constant, like \(10\), with respect to any variable is zero.
• Sum Rule: When differentiating a sum or difference of functions, differentiate each term separately. In this case, take the derivative of each function part independently before combining them.

So, applying these rules to the function \(D = 10 - \ln p\): The derivative is found by separately differentiating \(10\) (which results in 0) and \(- \ln p\) (which results in \(-\frac{1}{p}\)). Combining these, the final derivative simplifies to \(-\frac{1}{p}\). These rules make calculus tasks far more manageable.

Calculus problems

Calculus problems often involve finding derivatives, which can seem daunting without a clear approach. They usually require:

• Identifying the function and what needs to be differentiated.
• Applying the correct rules to break down the function into manageable parts.
• Simplifying the expression to reach the final result.

Let's take the example function \(D = 10 - \ln p\). By identifying the components—one constant and one logarithmic function—you use the rules correctly. The derivative \(-\frac{1}{p}\) becomes a straightforward expression, demonstrating that every calculus problem has a methodical solution path. Practice recognizing patterns and applying the right derivative rules to enhance problem-solving skills in calculus.