Question

Let f(x)=(x⁵+2)⁷/(sin⁴x cos³x) Find f'(x) using logarithmic differentiation.

Answer 1

To find the derivative of the function f(x) = (x⁵ + 2)⁷ / (sin⁴x cos³x) using logarithmic differentiation, follow the steps of taking the natural logarithm, simplifying the expression using logarithmic properties, differentiating using the chain rule and product rule, and simplifying the obtained derivative expression.

Step-by-step explanation:

To find the derivative of the function f(x) = (x⁵ + 2)⁷ / (sin⁴x cos³x) using logarithmic differentiation, we can follow these steps:

1. Take the natural logarithm of both sides of the equation: ln(f(x)) = ln((x⁵ + 2)⁷ / (sin⁴x cos³x)).
2. Apply logarithmic properties to simplify the expression. Use the fact that ln(a/b) = ln(a) - ln(b) and ln(aᵇ) = b ln(a).
3. Differentiate both sides of the equation with respect to x using the chain rule and product rule.
4. Solve for f'(x) by simplifying the obtained derivative expression.

After following these steps, you should obtain the derivative f'(x) = ... (final expression).