Differentiation of Constant – Proof and Examples

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

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