Question

If \[y = {e^x}\log x\] then \[\dfrac{{dy}}{{dx}}\] is
A. \[\dfrac{{{e^x}}}{x}\]
B. \[{e^x}\left( {\log x + \dfrac{1}{x}} \right)\]
C. \[{e^x}\left( {x\log x + \dfrac{1}{x}} \right)\]
D. \[\dfrac{{{e^x}}}{{\log x}}\]

Answer 1

Hint: The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Here in this question we will have to use the product rule (or Leibniz product rule).

Complete step by step answer:
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
In calculus, the product rule (or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as
\[\dfrac{d}{{dx}}\left( {u.v} \right) = \dfrac{{du}}{{dx}}.v + u.\dfrac{{dv}}{{dx}}\]
Where \[\dfrac{d}{{dx}}\left( {u.v} \right) = \] derivative of \[u.v\] with respect to \[x\] .
\[\dfrac{{du}}{{dx}}\] = derivative of \[u\] with respect to \[x\] .
\[\dfrac{{dv}}{{dx}}\] = derivative of \[v\] with respect to \[x\] .
Now consider the equation
\[y = {e^x}\log x\]
Using the product rule stated as above we have
\[\dfrac{{dy}}{{dx}} = {e^x}\log x + {e^x}\dfrac{1}{x}\]
Therefore we get
\[\dfrac{{dy}}{{dx}} = {e^x}\left( {\log x + \dfrac{1}{x}} \right)\]

So, the correct answer is “Option B”.

Note: Before finding the derivatives, we must always check if the function is continuous and differentiable at all points. To verify the obtained answer we can find the integration of the solution as integration is defined as the inverse of the derivation.