State the definition of the derivative from first principles.

Here are the derivatives of the given functions using first principles:

1) $f(x) = \cos x$

Step 1: State the definition of the derivative from first principles.

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Step 2: Substitute $f(x) = \cos x$ and $f(x+h) = \cos(x+h)$ into the formula.

$$f'(x) = \lim_{h \to 0} \frac{\cos(x+h) - \cos x}{h}$$

Step 3: Apply the sum-to-product trigonometric identity $\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$.

$$f'(x) = \lim_{h \to 0} \frac{-2 \sin\left(\frac{x+h+x}{2}\right) \sin\left(\frac{x+h-x}{2}\right)}{h}$$ $$f'(x) = \lim_{h \to 0} \frac{-2 \sin\left(x+\frac{h}{2}\right) \sin\left(\frac{h}{2}\right)}{h}$$

Step 4: Rearrange the expression to use the fundamental limit $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$.

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