Question

Find the derivative of y with respect to x if y = ${{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10$.
(a) $\dfrac{1}{x{{\log }_{e}}10}-\dfrac{{{\log }_{e}}10}{x{{\left( {{\log }_{e}}x \right)}^{2}}}$
(b) $\dfrac{1}{x{{\log }_{e}}10}-\dfrac{{{\log }_{e}}10}{x{{\log }_{10}}e}$
(c) $\dfrac{1}{x{{\log }_{e}}10}+\dfrac{{{\log }_{e}}10}{x{{\left( {{\log }_{e}}x \right)}^{2}}}$

Answer 1

Hint: As we can see, the equation y = ${{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10$ is a logarithmic function. Thus, to begin with, we will first understand what does the logarithmic operator represents in mathematics. Then, to find the derivative, we need to first simplify the equation of y. We will simplify it using the rules of logarithm and reduce the equation into differentiable forms. Then we will carry out the differentiation with respect to x.

Complete step-by-step solution:
A logarithmic function or log function is used when we have to deal with the powers of a number. To understand it better, we will see an example.
Suppose $a={{b}^{c}}$, where a, b and c are real numbers.
Then if we apply log on both sides with base b, we will get the following results.
$\Rightarrow {{\log }_{b}}a=c$
The log operator has many properties. Some of the properties which we will use in this particular problem are listed below:
${{\log }_{a}}a=1......\left( 1 \right)$
${{\log }_{a}}b=\dfrac{{{\log }_{e}}b}{{{\log }_{e}}a}......\left( 2 \right)$
The equation given to us is as follows:
y = ${{\log }_{10}}x+{{\log }_{x}}10+{{\log }_{x}}x+{{\log }_{10}}10$
With the help of (1), the equation modifies as follows:
y = ${{\log }_{10}}x+{{\log }_{x}}10$ + 1 + 1
With the help of (2), the equation modifies as follows:
y = $\dfrac{{{\log }_{e}}x}{{{\log }_{e}}10}+\dfrac{{{\log }_{e}}10}{{{\log }_{e}}x}$ + 2
${{\log }_{e}}$ is also written as ln.
Thus, y = $\dfrac{\ln x}{\ln 10}+\dfrac{\ln 10}{\ln x}$ + 2
The equation y = $\dfrac{\ln x}{\ln 10}+\dfrac{\ln 10}{\ln x}$ + 2 is differentiable function.
Therefore, we shall now differentiate y with respect to x. We shall keep in mind that the derivate of ln(x) is $\dfrac{1}{x}$. To find the derivative of $\dfrac{1}{\ln x}$, we will use concepts of differentiation of compound functions.
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{x\ln 10}-\dfrac{\ln 10}{x{{\left( \ln x \right)}^{2}}}$
Hence, option (a) is the correct option.

Note: Suppose we have a compound function given as $f\left( g\left( x \right) \right)$. Then differentiation of this function will be as follows: $\dfrac{df\left( g\left( x \right) \right)}{dx}=f'\left( g\left( x \right) \right)\times g'\left( x \right)$ . This is known as the chain rule of differentiation.