Find the Derivative - d/dx y=2^(3^(x^2))

Problem

d()/d(x)*2(3(x2))

Solution

1. Identify the function as a composition of exponential functions, requiring the chain rule and the rule for differentiating au which is d(au)/d(x)=au*ln(a)d(u)/d(x)

2. Apply the chain rule to the outermost base 2 treating 3(x2) as the exponent u

d(2(3(x2)))/d(x)=2(3(x2))*ln(2)d(3(x2))/d(x)

3. Differentiate the inner function 3(x2) by applying the chain rule again, this time treating x2 as the exponent.

d(3(x2))/d(x)=3(x2)*ln(3)d(x2)/d(x)

4. Differentiate the innermost power x2 using the power rule.

d(x2)/d(x)=2*x

5. Substitute the results back into the chain rule sequence.

d(2(3(x2)))/d(x)=2(3(x2))*ln(2)⋅3(x2)*ln(3)⋅2*x

6. Rearrange the terms to simplify the final expression.

d(2(3(x2)))/d(x)=2*x*ln(2)*ln(3)*2(3(x2))*3(x2)

Final Answer

d(2(3(x2)))/d(x)=2*x*ln(2)*ln(3)*2(3(x2))*3(x2)

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