Derivative of \( y = x^{x^{2x}} \) Explained Step-by-Step

This post will walk you through how to differentiate the complex exponential function:

Step 1: Apply Natural Logarithm

Take natural log on both sides to simplify:

\( \ln y = \ln\left(x^{x^{2x}}\right) = x^{2x} \cdot \ln x \)

Step 2: Differentiate Both Sides

Use implicit differentiation:

\( \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}(x^{2x} \cdot \ln x) \)

Step 3: Apply Product Rule

Let \( u = x^{2x} \), \( v = \ln x \). Then:

\( \frac{d}{dx}(uv) = u'v + uv' \)

Differentiate \( u = x^{2x} \):

Take log: \( \ln u = 2x \ln x \)
Then differentiate:
\( \frac{1}{u} \cdot \frac{du}{dx} = 2 \ln x + 2 \)
So,
\( \frac{du}{dx} = x^{2x}(2 \ln x + 2) \)

Differentiate \( v = \ln x \):

\( \frac{dv}{dx} = \frac{1}{x} \)

Step 4: Put It All Together

\[ \frac{1}{y} \cdot \frac{dy}{dx} = x^{2x}(2 \ln x + 2)\cdot \ln x + \frac{x^{2x}}{x} \]

Multiply both sides by \( y = x^{x^{2x}} \):

\[ \frac{dy}{dx} = x^{x^{2x}} \left[ x^{2x}(2 \ln x + 2)\cdot \ln x + \frac{x^{2x}}{x} \right] \]

Final Answer:

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Tags:

• derivative of x^x^2x
• logarithmic differentiation
• calculus step by step
• advanced derivative problems
• how to differentiate complex powers