Understanding Derivatives: The Derivative of a Constant is Always Zero

𝑑/𝑑𝑥[𝑐]

The expression 𝑑/𝑑𝑥[𝑐] represents the derivative of a constant value 𝑐 with respect to the variable 𝑥. When

The expression 𝑑/𝑑𝑥[𝑐] represents the derivative of a constant value 𝑐 with respect to the variable 𝑥. When you take the derivative of a constant, the result is always zero.

To see why this is the case, we can use the definition of the derivative. The derivative of a function 𝑓(𝑥) is defined as the limit of the difference quotient as the change in 𝑥 approaches zero:

𝑓'(𝑥) = lim (𝑓(𝑥 + 𝑘) – 𝑓(𝑥))/𝑘 as 𝑘→0.

Now, let’s apply this definition to the constant function 𝑐:

𝑐'(𝑥) = lim (𝑐 – 𝑐)/𝑘 as 𝑘→0.

Since 𝑐 – 𝑐 is equal to zero for any value of 𝑐, the numerator of the difference quotient is always zero:

𝑐'(𝑥) = lim (0)/𝑘 as 𝑘→0.

No matter what value 𝑘 approaches, the numerator remains zero while the denominator approaches zero. This means that the entire expression approaches zero:

𝑐'(𝑥) = 0.

Therefore, the derivative of a constant 𝑐 with respect to 𝑥 is always zero: 𝑑/𝑑𝑥[𝑐] = 0.

More Answers: