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

Problem

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

Solution

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

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

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

3. Differentiate the inner function 3^(x^2) by applying the chain rule again, this time treating x^2 as the exponent.

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

4. Differentiate the innermost power x^2 using the power rule.

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

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

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

6. Rearrange the terms to simplify the final expression.

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

Final Answer

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

Want more problems? Check here!