Question

What is the derivative of\[ - 5x\]?

Answer 1

Hint: Here we see that \[ - 5x\] is a linear function so the differentiation or the derivative of a linear function is linear. From previous knowledge, we know that the derivative of a linear function of the form \[mx + b\] with respect to \[x\] is m . Since the derivative of the constant function b is \[0\]. We also know that the derivative of \[a{x^n}\] is \[an{x^{n - 1}}\].

Complete step-by-step solution:
Given that the function is \[ - 5x\] , which is a linear function. We now use the formula of the derivative of \[a{x^n}\] is \[an{x^{n - 1}}\].
We observe that here \[a = - 5\]and \[n = 1\].
So, the derivative is \[an{x^{n - 1}}\] i.e. \[ = ( - 5).(1).{x^{1 - 1}}\]
\[ = - 5{x^0}\]
From our previous knowledge, we also know that \[{x^0}\] provided \[x \ne 0\] is always equal to $1$.
Then, \[ = ( - 5).(1)\]\[ = ( - 5)\]
Therefore, the derivative of \[ - 5x\] is \[ - 5\].

Note: It is very important that we know the basic derivative formulas of functions and that we recognize the function whether it is trigonometric, algebraic, logarithmic, etc, or the mixture of functions, and then proceed accordingly with the derivative formulas.
The derivative of a function \[y = f(x)\] of a variable \[x\] is a measure of the rate at which the value \[y\] of the function changes with respect to the change of the variable \[x\]. The derivative of a linear function \[mx + b\] can be derived using the definition of the derivative. The linear function derivative is a constant and is equal to the slope of the linear function i.e. \[m\]. Also, graphically the derivative of a function at a point can be defined as the instantaneous rate of change or as the slope of the tangent line to the graph of the function at the point. We can say that this slope of the tangent of a function at a point is the slope of the function.