What is the derivative of sin(x)?

Definition

The derivative of a function measures how the function's output changes as its input changes. For the sine function, $\sin(x)$, the derivative tells us the rate at which $\sin(x)$ changes with respect to $x$.

The derivative of $\sin(x)$ with respect to $x$ is:

$$ \frac{d}{dx} \sin(x) = \cos(x) $$

Worked Example

Let-s find the derivative of $f(x) = \sin(x)$ using the definition of the derivative:

The definition is:

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

Substitute $f(x) = \sin(x)$:

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

Using the sine addition formula:

$$ \sin(x+h) = \sin(x)\cos(h) + \cos(x)\sin(h) $$

So,

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

Group terms:

$$ = lim_{h \to 0} \left[ \sin(x) \frac{\cos(h) - 1}{h} + \cos(x) \frac{\sin(h)}{h} \right] $$

Take the limit:

• $lim_{h \to 0} \frac{\cos(h) - 1}{h} = 0$
• $lim_{h \to 0} \frac{\sin(h)}{h} = 1$

So,

$$ f-(x) = \sin(x) \cdot 0 + \cos(x) \cdot 1 = \cos(x) $$

Takeaways

• The derivative of $\sin(x)$ is $\cos(x)$.
• This result is fundamental in calculus and trigonometry.
• Knowing this derivative helps solve many problems involving rates of change in periodic phenomena.