\(\text{Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers)}\)\((px+q)(\frac{r}{x}+s).\)
\(\text{Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers)}\)\((px+q)(\frac{r}{x}+s).\)
Solution and Explanation
\(\text{Let }, f(x)=(px+q)(\frac{r}{x}+s)\)
\(\text{By Leibnitz product rule,}\)
\(=f'(x)=(px+q)(\frac{r}{x}+s)'+(\frac{r}{x}+s)(px+q)'\)
\(=(px+q)(rx^{-1}+s)'+(\frac{r}{s})(p)\)
\(=(px+q)(-rx^{-2})+(\frac{r}{s}+s)p\)
\(=(px+q)(\frac{-r}{x^2})+(\frac{r}{s})p\)
\(=-\frac{pr}{x}-\frac{qr}{x^2}+\frac{pr}{x}+ps\)
\(=ps-\frac{qr}{x^2}\)
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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