Question

If \[y = {5^x}{x^5}\], then \[\dfrac{{dy}}{{dx}}\] is equal to
A) \[{5^x}\left( {{x^5}\log 5 - 5{x^4}} \right)\]
B) \[{x^5}\log 5 - 5{x^4}\]
C) \[{x^5}\log 5 + 5{x^4}\]
D) \[{5^x}\left( {{x^5}\log 5 + 5{x^4}} \right)\]

Answer 1

Hint: Here, the given question. We have to find the derivative or differentiated term of the function. For this, first consider the given, then differentiate \[v\] with respect to \[x\] by using a standard differentiation formula and use product rule for differentiation then on further simplification we get the required differentiation value.

Complete step by step answer:
Differentiation can be defined as a derivative of a function with respect to an independent variable
(or)
The differentiation of a function is defined as the derivative or rate of change of a function. The function is said to be differentiable if the limit exists.
Let \[y = f\left( x \right)\] be a function of. Then, the rate of change of “y” per unit change in “x” is given by \[\dfrac{{dy}}{{dx}}\].
The product rule is used to differentiate many functions where one function is multiplied by another. The formal definition of the rule is: \[\dfrac{d}{{dx}}\left( {uv} \right) = u \cdot \dfrac{{dv}}{{dx}} + v \cdot \dfrac{{du}}{{dx}}\]
Consider the given function
 \[ \Rightarrow \,\,\,\,y = {5^x}{x^5}\]---------- (1)
Here, \[y\] is a dependent variable and \[x\] is an independent variable.
Now we have to differentiate this function with respect to \[x\]
\[ \Rightarrow \,\,\,\,\dfrac{d}{{dx}}\left( y \right) = \dfrac{d}{{dx}}\left( {{5^x}{x^5}} \right)\]
\[ \Rightarrow \,\,\,\,\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{5^x}{x^5}} \right)\]
Apply a product rule of differentiation, then
\[ \Rightarrow \,\,\,\,\dfrac{{dy}}{{dx}} = {5^x} \cdot \dfrac{d}{{dx}}\left( {{x^5}} \right) + \dfrac{d}{{dx}}\left( {{5^x}} \right) \cdot {x^5}\] -----(2)
Using the standard differentiated formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\] and \[\dfrac{d}{{dx}}\left( {{a^x}} \right) = {a^x}\log a\], then equation (2) becomes
\[ \Rightarrow \,\,\,\,\dfrac{{dy}}{{dx}} = {5^x} \cdot \left( {5{x^4}} \right) + \left( {{5^x}\log 5} \right) \cdot {x^5}\]
Take ‘\[{5^x}\]’ as common, then we have
\[ \Rightarrow \,\,\,\,\dfrac{{dy}}{{dx}} = {5^x}\left( {5{x^4} + \log 5 \cdot {x^5}} \right)\]
Or
\[\therefore \,\,\,\,\dfrac{{dy}}{{dx}} = {5^x}\left( {{x^5}\log 5 + 5{x^4}} \right)\]
Hence, the required differentiated value \[\dfrac{{dy}}{{dx}} = {5^x}\left( {{x^5}\log 5 + 5{x^4}} \right)\]. Therefore, option (D) is the correct answer.

Note:
When differentiating the function or term the student must recognize the dependent and independent variable then differentiate the dependent variable with respect to the independent variable. Should know the product and quotient rule of differentiation and remember the standard differentiation formulas.