Certainly! Let's find the derivative of the function \( y = x^{8x} \) using logarithmic differentiation.
Step 1: Take the natural logarithm of both sides.
We begin by taking the natural logarithm (\(\ln\)) of both sides of the equation:
[tex]\[
\ln(y) = \ln(x^{8x})
\][/tex]
Step 2: Simplify the right-hand side.
Using properties of logarithms, \(\ln(a^b) = b \cdot \ln(a)\), we can simplify the right-hand side of our equation:
[tex]\[
\ln(y) = 8x \cdot \ln(x)
\][/tex]
Step 3: Differentiate both sides with respect to \( x \).
Now we will differentiate both sides of the equation with respect to \( x \). Remember that \(\ln(y)\) on the left side requires us to use the chain rule:
[tex]\[
\frac{d}{dx}[\ln(y)] = \frac{d}{dx}[8x \cdot \ln(x)]
\][/tex]
The derivative of \(\ln(y)\) with respect to \( x \) is \(\frac{1}{y} \cdot \frac{dy}{dx}\). The derivative of \( 8x \cdot \ln(x) \) requires the product rule. So we get:
[tex]\[
\frac{1}{y} \cdot \frac{dy}{dx} = 8 \cdot \ln(x) + 8
\][/tex]
Step 4: Solve for \(\frac{dy}{dx}\).
Now we multiply both sides by \( y \) to isolate \(\frac{dy}{dx}\):
[tex]\[
\frac{dy}{dx} = y \cdot (8 \cdot \ln(x) + 8)
\][/tex]
Step 5: Substitute back the original function \( y = x^{8x} \).
We know that \( y = x^{8x} \). Substituting this back in for \( y \):
[tex]\[
\frac{dy}{dx} = x^{8x} \cdot (8 \cdot \ln(x) + 8)
\][/tex]
So, the derivative of the function \( y = x^{8x} \) is:
[tex]\[
y^{\prime}(x) = 8x^{8x} \left( \ln(x) + 1 \right)
\][/tex]
This is the final result for the derivative of the given function.
