Questions: Find the derivative of the function. f(t)=cos^2(e^cos^2(t)) f'(t)=

Solution Steps

To find the derivative of the function \( f(t) = \cos^2\left(e^{\cos^2(t)}\right) \), we will use the chain rule multiple times. First, identify the outer function and the inner functions. The outermost function is the square of cosine, followed by the exponential function, and then the cosine squared function. Differentiate each layer step by step, applying the chain rule at each stage.

We start with the function defined as: \[ f(t) = \cos^2\left(e^{\cos^2(t)}\right) \]

To find the derivative \( f'(t) \), we apply the chain rule. The derivative is computed as follows: \[ f'(t) = 4 \cdot \cos\left(e^{\cos^2(t)}\right) \cdot \sin\left(e^{\cos^2(t)}\right) \cdot e^{\cos^2(t)} \cdot \frac{d}{dt}\left(\cos^2(t)\right) \] where \( \frac{d}{dt}\left(\cos^2(t)\right) = -2\cos(t)\sin(t) \).

Substituting the derivative of \( \cos^2(t) \) into the expression, we have: \[ f'(t) = 4 \cdot e^{\cos^2(t)} \cdot \sin\left(e^{\cos^2(t)}\right) \cdot \cos\left(e^{\cos^2(t)}\right) \cdot (-2\cos(t)\sin(t)) \] This simplifies to: \[ f'(t) = -8 \cdot e^{\cos^2(t)} \cdot \sin\left(e^{\cos^2(t)}\right) \cdot \cos\left(e^{\cos^2(t)}\right) \cdot \cos(t) \cdot \sin(t) \]

Final Answer