Find the derivative of the following function. $$ y=x^{5}\left(4-\ln x^{5}\right) $$ $$ \frac{d y}{d x}=\square $$

Question

UpStudy AI · Accepted Answer

We start with the function

y = x⁵(4 − ln(x⁵)).

To differentiate, we use the product rule. Let

f(x) = x⁵  and  g(x) = 4 − ln(x⁵).

The product rule states that

y′ = f′(x)·g(x) + f(x)·g′(x).

Step 1. Differentiate f(x):

f(x) = x⁵  ⟹  f′(x) = 5x⁴.

Step 2. Differentiate g(x):

g(x) = 4 − ln(x⁵).

The derivative of 4 is 0. For ln(x⁵), use the chain rule. Recall that

d/dx [ln(u)] = (1/u)·(du/dx).

Here, u(x) = x⁵, so du/dx = 5x⁴. Then:

d/dx [ln(x⁵)] = (1/(x⁵))·(5x⁴) = 5/x.

Thus:

g′(x) = 0 − (5/x) = −5/x.

Step 3. Apply the product rule:

y′ = 5x⁴(4 − ln(x⁵)) + x⁵(−5/x).

Simplify the second term:

x⁵(−5/x) = −5x⁴.

Now, we have:

y′ = 5x⁴(4 − ln(x⁵)) − 5x⁴.

Factor out 5x⁴:

y′ = 5x⁴[(4 − ln(x⁵)) − 1] = 5x⁴(3 − ln(x⁵)).

Thus, the derivative is:

dy/dx = 5x⁴(3 − ln(x⁵)).
The derivative is $$ \frac{dy}{dx} = 5x^{4}(3 - \ln x^{5}) $$.

Find the derivative of the following function. \( y=x^{5}\left(4-\ln x^{5}\right) \) \( \frac{d y}{d x}=\square \)