Here you will learn the differentiation of constant function proof and examples.
Let’s begin –
Differentiation of Constant
The differentiation of constant function is zero. i.e. \(d\over dx\)(c) = 0.
Proof : Let f(x) = c, be a constant function. Then,
By using first principle,
\(d\over dx\) (f(x)) = \(\displaystyle{\lim_{h \to 0}}\) \(f(x + h) – f(x)\over h\)
= \(\displaystyle{\lim_{h \to 0}}\) \(c – c\over h\) = 0
Hence, \(d\over dx\)(c) = 0, where c is a constant.
Remark : Geometrically, graph of a constant function is a straight line parallel to x-axis. So, tangent at every point is parallel to x-axis. Consequently slope of the tangent is zero, i.e. \(dy\over dx\) = 0.
Also Read : Differentiation Formulas Class 12
Note : Let f(x) be a differentiable function and let c be a constant. Then c.f(x) is also differentiable such that
\(d\over dx\){c.f(x)} = c.\(d\over dx\)(f(x))
i.e. the derivative of a constant times a function is the constant times the derivatives of the function.
Example : Differentiate the following with respect to x.
(i) 5
(ii) 5x
(iii) \(log_x x\)
Solution :
(i) we have,
f(x) = 5, which is a constant.
Therefore, \(d\over dx\)(f(x)) = \(d\over dx\) (5) = 0
(ii) we have,
f(x) = 5x
Therefore, \(d\over dx\)(f(x)) = \(d\over dx\) (5x) = 5\(d\over dx\) (5) = 5(1) = 5
(iii) we have,
f(x) = \(log_x x\) = 1
Therefore, \(d\over dx\)(f(x)) = \(d\over dx\) (1) = 0