Answer:
[tex]\frac{dy}{dx} = \frac{4(In x)^{3}}{x}[/tex]
Step-by-step explanation:
Solving [tex]y = (In)^{4}[/tex]
Using function of function (Chain Rule)
[tex]\frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx} [/tex] -----(1)
To simplify the function,
Let u = In x
Differentiating u with respect to x
[tex]\frac{du}{dx} = \frac{1}{x}[/tex] -------- (2)
Substituting u = In x in the given function [tex]y = (In x)^{4}[/tex]
[tex]y = (u)^{4}[/tex]
Differentiating y with respect to u
[tex]\frac{dy}{du} = 4u^{3}[/tex]
Now that we have [tex]\frac{dy}{du}[/tex] and [tex]\frac{du}{dx}[/tex], we can solve for [tex]\frac{dy}{dx}[/tex]
[tex]\frac{dy}{dx} = 4u^{3} * \frac{1}{x}[/tex]
Substituting u = In x into the equation,
[tex]\frac{dy}{dx} = 4(In x)^{3} * \frac{1}{x}[/tex]
[tex]\frac{dy}{dx} = \frac{4(In x)^{3}}{x}[/tex]
