Final answer:
Logarithmic differentiation is used to find the derivative of y = x^(5^x). By taking the natural logarithm of both sides and applying the logarithm properties, we differentiate implicitly to find dy/dx. The final answer is dy/dx = x^(5^x) · [(5^x · (1/x)) + (ln(x) · 5^x · ln(5))].
Step-by-step explanation:
Logarithmic Differentiation of y = x^(5^x)
To find the derivative of the function y = x^(5^x) using logarithmic differentiation, we first take the natural logarithm of both sides:
ln(y) = ln(x^(5^x))
Applying the logarithm property of exponents, we move the variable exponent in front:
ln(y) = 5^x · ln(x)
Now differentiate implicitly with respect to x. Remember to use the product rule on the right side:
(1/y) · (dy/dx) = 5^x · (1/x) + ln(x) · d(5^x)/dx
The derivative of 5^x with respect to x involves another application of logarithmic differentiation:
d(5^x)/dx = 5^x · ln(5)
Substitute back into our original equation:
(1/y) · (dy/dx) = 5^x · (1/x) + ln(x) · 5^x · ln(5)
Multiply both sides by y to solve for dy/dx:
dy/dx = y · [5^x · (1/x) + ln(x) · 5^x · ln(5)]
Recalling that y = x^(5^x), we have our final answer:
dy/dx = x^(5^x) · [(5^x · (1/x)) + (ln(x) · 5^x · ln(5))]