Find the derivative of the following functions (it is to be understood that \(a, \,b,\, c,\, d,\, p,\, q,\, r\) and \(s\) are fixed non-zero constants and \(m\) and \(n\) are integers), \((x+a)\).
Find the derivative of the following functions (it is to be understood that \(a, \,b,\, c,\, d,\, p,\, q,\, r\) and \(s\) are fixed non-zero constants and \(m\) and \(n\) are integers), \((x+a)\).
Solution and Explanation
\(\text{Let f(x) = x + a. Accordingly } f(x+h)=x+h+a\)
\(\text{By first principle,}\)
\(f'(x)=\)\(\frac{f(x+h)-f(x)}{h}\)
\(=\underset{h→0}{lim} \,\frac{x+h+-x-a}{h}\)
\(= \underset{h→0}{lim} \,(\frac{h}{h})\)
\(=\underset{h→0}{lim} \,(1)\)
\(=1\)
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Concepts Used:
Limits And Derivatives
Mathematically, a limit is explained as a value that a function approaches as the input, and it produces some value. Limits are essential in calculus and mathematical analysis and are used to define derivatives, integrals, and continuity.
Limits Formula:
Derivatives of a Function:
A derivative is referred to the instantaneous rate of change of a quantity with response to the other. It helps to look into the moment-by-moment nature of an amount. The derivative of a function is shown in the below-given formula.
Properties of Derivatives:
Read More: Limits and Derivatives