Question

Can derivatives be zero?

Answer 1

To solve this question we need to have the knowledge of derivatives. Derivative is defined as the rate of change of a function with respect to a variable. We will solve the question using some of the examples so that we get a clear idea of the derivatives for the functions assigned to us.

Complete step-by-step solution:
The question asks us whether the derivatives of a function can be zero or not. To start with the definition of the derivative. Derivative is defined as the rate of change of a function given to us with respect to any variable. We will check whether the function given to us is a dependent variable or not. If a derivative of the function given to us is found with respect to one of the independent variables of the function then the derivative will be a real number. For example if we consider the function $f=x+5y+3z$, in this case the function $f$ is dependent upon $x,y,z$ variables. Now if we find the derivative of the function with respect to $x$ we will get:
$\Rightarrow \dfrac{d(x+5y+3z)}{dx}$
$\Rightarrow 1$
Similarly on finding the derivative of the function with respect to $y$ we will get:
$\Rightarrow \dfrac{d(x+5y+3z)}{dy}$
$\Rightarrow 5$
Similarly on finding the derivative of the function with respect to $z$ we will get:
$\Rightarrow \dfrac{d(x+5y+3z)}{dz}$
$\Rightarrow 3$
From the above three derivatives we see that, if the derivative of the function is found with respect to a variable on which the function is dependent then the derivative will be non- zero value.
If we find the derivative of the function with respect to another function or any other variable other than that on which the function is not dependent then the derivative is zero. For instance if the above function is differentiated with respect to any other variable other than $x,y,z$, let's say on differentiating with $a$, then we get:
$\Rightarrow \dfrac{d(x+5y+3z)}{da}$
$\Rightarrow 0$
So for the function which is constant with respect to the variable with respect to which change is found, the derivative will be zero.
$\therefore$ The derivatives can be zero.

Note: If the derivative of the function is given as zero then the function is a constant value which could be found using integration, to be more precise by using definite integration. If $\dfrac{df}{dx}=0$ , then $df=0.dx$ . On integrating we get $\int{df=\int{0.dx}}$, which will further result in $f=c$, where $c$ is the constant.