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The derivative of a function provides us with the rate at which the function changes at any given point. In simple terms, it tells us how steep the curve of the function is at a particular spot.
To compute the derivative of a function at a point, we use a formula involving limits. This formula is known as the definition of the derivative. It is expressed as:
\(\frac{d}{dx}f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)
In this formula, \(h\) symbolizes a small change in the \(x\) value. If we think about \(x+h\), it means we move slightly away from \(x\).
Taking the limit as \(h\) approaches zero means looking at what happens as the change in \(x\) gets smaller and smaller. This way, we find the instantaneous rate of change at \(x\). For our specific example, we are dealing with \(\cos(x)\), and our goal is to find how \(\cos(x)\) changes as \(x\) changes.

Trigonometric identities are equations involving trigonometric functions that are always true for any angle. These identities help simplify expressions and solve trigonometric equations.
One of the most useful identities in calculus is the angle addition formula for cosine:
\(\cos(x+h) = \cos x \cos h - \sin x \sin h\).
This identity lets us break down complex trigonometric expressions into simpler parts that are easier to work with.
In the context of derivatives, using these identities allows us to rearrange and manipulate functions to apply the limit definition effectively.
By substituting the identity into the derivative of \(\cos(x)\), we can easily see how the pieces fit together and apply limits to find the derivative.

The limit of a function describes what happens when we get closer and closer to a particular point on the function. In calculus, limits are foundational because they help us understand behavior near specific points.
Limits are expressed as:
\(\lim_{h \to 0} f(h)\)
This notation means we are interested in what value \(f(h)\) approaches as \(h\) becomes incredibly small.
For instance, in the context of derivatives, we often come across limits like \(\lim_{h \to 0} \frac{\cos h - 1}{h} = 0\) and \(\lim_{h \to 0} \frac{\sin h}{h} = 1\).
These specific limits are crucial in finding the derivative of trigonometric functions, letting us simplify expressions by knowing their behavior as \(h\) approaches zero.
Through limits, we gain a precise mathematical way to describe rates of change and instantaneous slopes, crucial for understanding and working with derivatives effectively.