Question

How do you find the first and second derivative of $\ln \left( {\dfrac{x}{{20}}} \right)$?

Answer 1

Here in this question, the given function is a logarithm function. Here we have to find the first and second derivative of the given logarithm function. First derivative can be find by using the logarithm property of Quotient rule i.e., ${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}m - {\log _b}n$ and differentiate the ln function and to find the second derivative again differentiate the answer of first derivative by using standard formula of differentiation.

Complete step-by-step solution:
Differentiation is the algebraic method of finding the derivative for a function at any point. The derivative is a concept that is at the root of calculus. Either way, both the slope and the instantaneous rate of change are equivalent, and the function to find both of these at any point is called the derivative. It can be represented as $\dfrac{{dy}}{{dx}}$ or $y'$, where y is the function of x.

A second-order derivative is a derivative of the derivative of a function. It is drawn from the first-order derivative. So we first find the derivative of a function and then draw out the derivative of the first derivative. It can be represented as $\dfrac{{{d^2}y}}{{d{x^2}}}$ or $y''$.

Consider the given logarithm function
$\Rightarrow \,\,\,\,y = \ln \left( {\dfrac{x}{{20}}} \right)$
Using the logarithm property of quotient rule i.e., ${\log _b}\left( {\dfrac{m}{n}} \right) = {\log _b}m - {\log _b}n$, the above equation can be written as
$\Rightarrow \,\,\,\,y = \ln \left( {\dfrac{x}{{20}}} \right) = \ln \left( x \right) - \ln \left( {20} \right)$
Now differentiate the equation with respect to $x$.
$\Rightarrow \,\,\,\,\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\ln x} \right) - \dfrac{d}{{dx}}\left( {\ln 20} \right)$
$\Rightarrow \,\,\,\,\dfrac{{dy}}{{dx}} = \dfrac{1}{x} - 0$
$\therefore \,\,\,\dfrac{{dy}}{{dx}} = \dfrac{1}{x}$
Or
$\therefore \,\,\,y' = \dfrac{1}{x}$
The first derivative of given function $\ln \left( {\dfrac{x}{{20}}} \right)$ is $\dfrac{{dy}}{{dx}} = \dfrac{1}{x}$.
To find the second derivative of $\ln \left( {\dfrac{x}{{20}}} \right)$
Consider the first derivative
$\Rightarrow \,\,\,\dfrac{{dy}}{{dx}} = \dfrac{1}{x}$
Again differentiate this equation with respect to $x$ using the formula of $\dfrac{{dy}}{{dx}}{x^n} = n{x^{n - 1}}$ or $\dfrac{{dy}}{{dx}}\left( {\dfrac{1}{x}} \right) = - \dfrac{1}{{{x^2}}}$, then
$\Rightarrow \,\,\,\dfrac{{{d^2}y}}{{d{x^2}}} = \dfrac{d}{{dx}}\left( {\dfrac{1}{x}} \right)$
$\therefore \,\,\,\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{{x^2}}}$
Or
$\therefore \,\,\,y'' = - \dfrac{1}{{{x^2}}}$

Hence the second derivative of given function $\ln \left( {\dfrac{x}{{20}}} \right)$ is $\dfrac{{{d^2}y}}{{d{x^2}}} = - \dfrac{1}{{{x^2}}}$.

Note: To differentiate or to find the derivative of a function we use some standard differentiation formulas. The derivative is the rate of change of quantity, in this question we differentiate the given function with respect to x and find the derivative. For differentiation we must know the standard differentiation formulas.