Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

Answer 1

Answer:

$\frac{dy}{dx} = \frac{4(In x)^{3}}{x}$

Step-by-step explanation:

Solving $y = (In)^{4}$

Using function of function (Chain Rule)

             $\frac{dy}{dx} = \frac{dy}{du} * \frac{du}{dx}$    -----(1)

To simplify the function,

             Let u = In x

Differentiating u with respect to x

             $\frac{du}{dx} = \frac{1}{x}$                -------- (2)

Substituting u = In x in the given function $y = (In x)^{4}$

             $y = (u)^{4}$

Differentiating y with respect to u

             $\frac{dy}{du} = 4u^{3}$

Now that we have $\frac{dy}{du}$ and $\frac{du}{dx}$, we can solve for $\frac{dy}{dx}$

             $\frac{dy}{dx} = 4u^{3} * \frac{1}{x}$

Substituting u = In x into the equation,

             $\frac{dy}{dx} = 4(In x)^{3} * \frac{1}{x}$

             $\frac{dy}{dx} = \frac{4(In x)^{3}}{x}$