To find the derivative of the function [tex]\( f(x) = \cos^2(5-x) \)[/tex] and then evaluate it at [tex]\( x = 0 \)[/tex], follow these steps:
1. Function Definition:
[tex]\[
f(x) = \cos^2(5 - x)
\][/tex]
2. Apply the Chain Rule:
To differentiate [tex]\( f(x) \)[/tex], we need to use the chain rule. Notice that [tex]\( f(x) = [\cos(5 - x)]^2 \)[/tex]. Let [tex]\( u = \cos(5 - x) \)[/tex]. Then [tex]\( f(x) = u^2 \)[/tex].
3. Differentiate [tex]\( u^2 \)[/tex] with respect to [tex]\( u \)[/tex]:
[tex]\[
\frac{d}{du}(u^2) = 2u
\][/tex]
4. Differentiate [tex]\( \cos(5 - x) \)[/tex] with respect to [tex]\( x \)[/tex]:
Using the chain rule again, let [tex]\( g(x) = 5 - x \)[/tex]. Then we have [tex]\( h(g) = \cos(g) \)[/tex], and [tex]\( g(x) = 5 - x \)[/tex]. Therefore,
[tex]\[
\frac{d}{dx}[\cos(5 - x)] = -\sin(5 - x)
\][/tex]
5. Combine the derivatives using the chain rule:
[tex]\[
\frac{d}{dx}[\cos^2(5 - x)] = 2\cos(5 - x) \cdot \left(-\sin(5 - x)\right)
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{d}{dx}[\cos^2(5 - x)] = -2\cos(5 - x) \sin(5 - x)
\][/tex]
6. Evaluate the derivative at [tex]\( x = 0 \)[/tex]:
[tex]\[
f'(x) = -2\cos(5 - x) \sin(5 - x)
\][/tex]
Substituting [tex]\( x = 0 \)[/tex]:
[tex]\[
f'(0) = -2\cos(5 - 0) \sin(5 - 0)
\][/tex]
[tex]\[
f'(0) = -2\cos(5) \sin(5)
\][/tex]
So, the derivative [tex]\( f'(0) = -2 \sin(5) \cos(5) \)[/tex]. From the given options, this matches the third option:
[tex]\[ 2 \cos (5) \sin (5) \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{2 \cos (5) \sin (5)} \][/tex]
