If $f ( x ) = 2 \sin ^ { 2 } x$ and $g ( x ) = - 3 \cos ^ { 2 } x$ , is $f ^ { \prime } ( x ) = g ^ { \prime } ( x )$ ?

Question

Quizplus · Accepted Answer

The derivatives of $f(x)$ and $g(x)$ involve different trigonometric functions and coefficients. Specifically, $f'(x) = 4\sin(x)\cos(x)$ using the chain rule and trigonometric identity $\sin^2(x) + \cos^2(x) = 1$, and $g'(x) = 6\sin(x)\cos(x)$, also using the chain rule and the same trigonometric identity. The coefficients 4 and 6 in the derivatives indicate that $f'(x)$ and $g'(x)$ are not equal.

If f(x)=2sin⁡2xf ( x ) = 2 \sin ^ { 2 } xf(x)=2sin2x

If $f ( x ) = 2 \sin ^ { 2 } x$