Question

Show that the derivative of a constant function on an interval is zero.

Answer 1

Hint: In the given question, we are required to evaluate the derivative of a constant function on an interval. So, we have to show that differentiation of a constant function on an interval yields zero. Now, we first assume a constant function and then try to find its derivative using the basic definition of derivative. Derivative of a function gives the rate of change of that particular function.

Complete step by step answer:
Differentiation provides us with the rate of change of a variable with respect to some other variable. Derivative of a function at a point gives the instantaneous slope of the function at that particular point. The slope of function describes the steepness of a straight line. Slope is also denoted by the symbol ‘$m$’.

Now, we know that a constant function always gives the same output for every input value of a variable. So, there is no change in the slope of a constant function. Hence, the derivative of a constant function on an interval is zero. For example: Let $f\left(x\right)=5$. Then, as we can see in the graph of the function, for every real value of x, the value of the function remains $5$.

Now, we see that the slope of the constant function $f\left(x\right)=5$ remains the same throughout the real number line.

Now, we know that the derivative of a function represents the slope of the function at a particular point. So, the derivative of the constant function over an interval is zero.

Note:We must know the basics of differentiation and limits in order to solve the given question. We can also calculate the derivative of a constant function by using the power rule of differentiation ${\displaystyle \frac{d\left(x^n\right)}{dx}}=n{x}^{\left(n-1\right)}$. We should always remember the result of this problem as it can be used in many other complex problems directly.