Compute the derivative. Use logarithmic differentiation where appropriate.
StartFraction d Over dx EndFraction left parenthesis 8 x right parenthesis Superscript 5 x
d
dx(8x)5x

We are tasked with finding the derivative of \( (8x)^{5x} \). Since both the base and the exponent contain \( x \), logarithmic differentiation is a useful method.

### Step 1: Take the natural logarithm of both sides
Let \( y = (8x)^{5x} \).

Taking the natural logarithm of both sides:

\[
\ln y = \ln \left( (8x)^{5x} \right)
\]

Using the logarithmic identity \( \ln(a^b) = b\ln(a) \), we get:

\[
\ln y = 5x \cdot \ln(8x)
\]

### Step 2: Differentiate both sides with respect to \( x \)
Now, differentiate both sides using implicit differentiation. On the left-hand side:

\[
\frac{d}{dx} \ln y = \frac{1}{y} \cdot \frac{dy}{dx}
\]

On the right-hand side, we apply the product rule to \( 5x \cdot \ln(8x) \). The product rule states that \( \frac{d}{dx}(uv) = u'v + uv' \). Here, \( u = 5x \) and \( v = \ln(8x) \).

We first find the derivatives of \( u \) and \( v \):
- \( u = 5x \) so \( u' = 5 \)
- \( v = \ln(8x) = \ln 8 + \ln x \), so \( v' = \frac{1}{x} \)

Now applying the product rule:

\[
\frac{d}{dx} \left( 5x \ln(8x) \right) = 5 \ln(8x) + 5x \cdot \frac{1}{x} = 5 \ln(8x) + 5
\]

Thus, the derivative of the right-hand side is:

\[
\frac{d}{dx} \ln y = 5 \ln(8x) + 5
\]

### Step 3: Solve for \( \frac{dy}{dx} \)
Now, substitute \( \frac{d}{dx} \ln y \) into the equation:

\[
\frac{1}{y} \cdot \frac{dy}{dx} = 5 \ln(8x) + 5
\]

Multiplying both sides by \( y = (8x)^{5x} \):

\[
\frac{dy}{dx} = (8x)^{5x} \left( 5 \ln(8x) + 5 \right)
\]

Thus, the derivative is:

\[
\frac{d}{dx} \left( (8x)^{5x} \right) = (8x)^{5x} \left( 5 \ln(8x) + 5 \right)
\]

This is the final answer.

Would you like further details or have any questions?

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Here are 5 related questions:
1. How does logarithmic differentiation work for functions like \( x^x \)?
2. What is the product rule in differentiation and when is it applied?
3. How do you differentiate logarithmic functions such as \( \ln(8x) \)?
4. Can logarithmic differentiation be used for non-exponential functions?
5. What are some other methods to differentiate complex exponential functions?

**Tip:** Logarithmic differentiation is particularly useful when both the base and the exponent contain variables.

Compute the derivative. Use logarithmic differentiation where appropriate. d/dx((8x)^5x)

Learn how to compute the derivative of (8x)^5x using logarithmic differentiation. This solution breaks down key steps, including applying the product rule and logarithmic identities. Suitable for calculus students.

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