The derivative of y = (x³ + 2)²(x⁴ + 4)⁴ using logarithmic differentiation is;
y' = {[6x²In(x³ + 2)]/(x³ + 2)} + {[16x³In(x⁴ + 4)⁴]/(x⁴ + 4)} × [(x³ + 2)²(x⁴ + 4)⁴]
We are given the function;
y = (x³ + 2)²(x⁴ + 4)⁴
We want to find the derivative using logarithmic differentiation;
Step 1; Take the natural log of both sides;
In y = In[(x³ + 2)²(x⁴ + 4)⁴]
Step 2;
Using log of a product property on this, we have;
In y = In(x³ + 2)² + In(x⁴ + 4)⁴
Step 3; We will now differentiate both sides with chain rule to get;.
y'/y = [6x²In(x³ + 2)]/(x³ + 2) + [16x³In(x⁴ + 4)⁴]/(x⁴ + 4)
Step 4; Using multiplication property of equality, multiply both sides by y to get;
y' = {[6x²In(x³ + 2)]/(x³ + 2) + [16x³In(x⁴ + 4)⁴]/(x⁴ + 4)} × y
Step 5; Plug in the value of y = (x³ + 2)²(x⁴ + 4)⁴ to get;
y' = {[6x²In(x³ + 2)]/(x³ + 2)} + {[16x³In(x⁴ + 4)⁴]/(x⁴ + 4)} × [(x³ + 2)²(x⁴ + 4)⁴]
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