Method of Differentiation
Logarithmic Differentiation
Premium Question
Grade 12
Question:
If $y = (\sin x)^{\cos x}$, then $\frac{dy}{dx}$ equals:
(1) $(\sin x)^{\cos x} [\cos x \cot x - \sin x \ln(\sin x)]$
(2) $(\sin x)^{\cos x} [\cos x \cot x + \sin x \ln(\sin x)]$
(3) $(\sin x)^{\cos x} [-\sin x \ln(\sin x)]$
(4) $(\sin x)^{\cos x} \cos x \cot x$
Step-by-Step Solution
Key Concept: Take logarithms of both sides to write $\ln y = \cos x \ln(\sin x)$, which turns the variable-exponent expression into a product that is straightforward to differentiate implicitly.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)