**Core Concept**
First-order kinetics refers to the process where the rate of elimination of a drug is directly proportional to its plasma concentration. This means that as the plasma concentration of the drug decreases, the rate of elimination also decreases, resulting in a linear relationship between drug concentration and elimination rate.

**Why the Correct Answer is Right**
The correct answer is related to the concept of first-order kinetics, where the rate of elimination is directly proportional to the plasma concentration of the drug. This is in contrast to zero-order kinetics, where the rate of elimination is constant and independent of plasma concentration. In first-order kinetics, the rate of elimination is determined by the enzyme responsible for the elimination process, such as cytochrome P450 in the liver. The enzyme's activity is affected by the plasma concentration of the drug, resulting in a linear relationship between drug concentration and elimination rate.

**Why Each Wrong Option is Incorrect**
**Option A:** This statement is incorrect because first-order kinetics is characterized by a linear relationship between drug concentration and elimination rate, not a constant rate of elimination.

**Option B:** This statement is incorrect because first-order kinetics is directly proportional to the plasma concentration of the drug, not inversely proportional.

**Option D:** This statement is incorrect because first-order kinetics is characterized by a decrease in the rate of elimination as the plasma concentration of the drug decreases, not an increase.

**Clinical Pearl / High-Yield Fact**
A key point to remember is that first-order kinetics is typically observed with drugs that are metabolized by enzymes, such as cytochrome P450 in the liver. This is in contrast to zero-order kinetics, which is often observed with drugs that are eliminated through non-enzymatic processes, such as renal excretion.

**Correct Answer: C. The rate of elimination of a drug is directly proportional to its plasma concentration.**