The derivative of \( -7 \) is \( 0 \).

["Title: The Derivative of a Constant: Why Is It Zero?", "Understanding derivatives is fundamental in calculus, especially when exploring how functions behave. A common question arises among students: What is the derivative of −7? At first glance, this might seem simple—or even confusing—but it reveals key principles about derivatives.", "### What Is a Derivative?", "The derivative of a function at a point represents the instantaneous rate of change of the function at that point. For example, if a function describes position over time, its derivative gives velocity—the speed at a specific moment.", "Mathematically, the derivative of a function ( f(x) ) at ( x = a ) is defined as:", "[\nf'(a) = \lim_{h \ o 0} \frac{f(a + h) - f(a)}{h}\n]", "This limit measures how ( f ) changes as ( x ) changes, but only over an infinitesimally small interval.", "### Applying It to a Constant Function", "Consider the constant function ( f(x) = -7 ). The output never changes—it remains exactly (-7) for any input ( x ). When you plug this into the derivative formula:", "[\nf'(x) = \lim_{h \ o 0} \frac{-7 - (-7)}{h} = \lim_{h \ o 0} \frac{0}{h} = \lim_{h \ o 0} 0 = 0\n]", "Because the numerator is always zero, no matter how small ( h ) gets, the difference quotient becomes zero. This makes intuitive sense: a constant function does not increase or decrease. Its slope is flat, and thus its derivative is zero everywhere.", "### Why Is the Derivative Zero?", "The derivative of a constant is always zero because a constant function has no change as ( x ) varies. There is no tendency to increase or decrease—no “momentum” in ( f(x) = -7 ). This applies universally: whether the constant is ( -7 ), ( \pi ), or ( 0 ), the derivative is ( 0 ).", "### Conclusion", "So, why is the derivative of ( -7 ) equal to ( 0 )? The reason lies in the definition of the derivative and the nature of constants. A constant function never changes, meaning its instantaneous rate of change is zero at every point. Understanding this builds a solid foundation in calculus and prepares learners for more complex functions.", "Remember: if ( f(x) = c ), a constant, then ( f'(x) = 0 ). In our case, ( c = -7 ), so ( f'(-7) = 0 ).", "---", "Keywords: derivative of −7, meaning of derivative, constant function derivative, calculus basics, derivative of a constant, why derivative is zero, derivative definition, instantaneous rate of change.\nMeta Description: Learn why the derivative of the constant −7 is 0. Explore calculus fundamentals and understand how constant functions behave in differentiation."]