Question

What is the derivative of y = ln(sin²(x)sec³(x)/(3x²)⁵) using logarithmic expansion laws?

A) d/dx[ln(sin²(x)sec³(x)/(3x²)⁵)]
B) 2cot(x)sec⁴(x) - 15/x
C) -10cot(x)sec⁴(x) - 15/x
D) d/dx[ln(sin²(x)sec³(x))] - d/dx[ln(3x²)⁵]

Answer 1

To find the derivative of y = ln(sin²(x)sec³(x)/(3x²)⁵) using logarithmic expansion laws, apply the expansion laws for division and multiplication, simplify the expression, and take the derivative. The result is -10cot(x)sec⁴(x) - 15/x.

Step-by-step explanation:

To find the derivative of y = ln(sin²(x)sec³(x)/(3x²)⁵), we need to apply the logarithmic expansion laws. Here's how:

1. Apply the logarithmic expansion law for division: ln(a/b) = ln(a) - ln(b). Apply this law to the expression inside the natural logarithm.
2. Apply the logarithmic expansion law for multiplication: ln(a*b) = ln(a) + ln(b). Apply this law to the expression inside the natural logarithm again.
3. Simplify the expression inside the natural logarithm.
4. Take the derivative of the simplified expression.

The final result of the derivative will be: -10cot(x)sec⁴(x) - 15/x.