To find the derivative of the function y = x^3 + 22x^4 + 44 using logarithmic differentiation, we start by taking the natural logarithm of both sides:
ln(y) = ln(x^3 + 22x^4 + 44)
Next, we can differentiate both sides with respect to x. Remember to apply the chain rule on the left side:
1/y * (dy/dx) = 1/(x^3 + 22x^4 + 44) * (3x^2 + 88x^3)
Now, multiply both sides by y to isolate dy/dx:
dy/dx = y * (3x^2 + 88x^3)/(x^3 + 22x^4 + 44)
We know that y = x^3 + 22x^4 + 44, so we can substitute back:
dy/dx = (x^3 + 22x^4 + 44) * (3x^2 + 88x^3)/(x^3 + 22x^4 + 44)
Finally, simplifying gives us:
dy/dx = 3x^2 + 88x^3
This is the derivative of the function using logarithmic differentiation!