If $f(x)=\sin ^4 x$, find $f^{\prime}(x)$
Answer (1)
Apply the chain rule to f ( x ) = sin 4 x .
Let u = sin x , so f ( x ) = u 4 .
Find d x d u = cos x and d u df = 4 u 3 .
Apply the chain rule: f ′ ( x ) = 4 sin 3 x cos x .
4 sin 3 x cos x
Explanation
Problem Analysis We are given the function f ( x ) = sin 4 x and we want to find its derivative, f ′ ( x ) . We will use the chain rule to differentiate this function.
Setting up the Chain Rule Let u = sin x . Then f ( x ) = u 4 . We need to find d x d u and d u df .
Finding du/dx First, we find the derivative of u with respect to x :
d x d u = d x d ( sin x ) = cos x
Finding df/du Next, we find the derivative of f with respect to u :
d u df = d u d ( u 4 ) = 4 u 3
Applying the Chain Rule Now, we apply the chain rule: d x df = d u df ⋅ d x d u = 4 u 3 ⋅ cos x
Substituting Back Substitute u = sin x back into the expression: d x df = 4 ( sin x ) 3 ⋅ cos x = 4 sin 3 x cos x
Final Answer Therefore, the derivative of f ( x ) = sin 4 x is: f ′ ( x ) = 4 sin 3 x cos x
Examples
Understanding derivatives of trigonometric functions is crucial in physics, especially when dealing with oscillatory motion. For instance, if f ( x ) = sin 4 x represents the power of a signal varying with time x , then f ′ ( x ) = 4 sin 3 x cos x describes how the power changes instantaneously. This is essential in analyzing signal strength and optimizing communication systems.
