The derivatives of sine and cosine follow a four-step cycle:
$\cos x \to -\sin x \to -\cos x \to \sin x \to \cos x$.
Knowing this cycle allows you to find any $n^{th}$ order derivative quickly!
Step 1: Find the first derivative ($\frac{dy}{dx}$): The derivative of $\cos x$ with respect to $x$ is:
$$\frac{dy}{dx} = -\sin x$$
Step 2: Find the second derivative ($\frac{d^2y}{dx^2}$): Differentiate the first derivative again with respect to $x$:
$$\frac{d^2y}{dx^2} = \frac{d}{dx}(-\sin x)$$
The derivative of $\sin x$ is $\cos x$. Therefore:
$$\frac{d^2y}{dx^2} = -(\cos x) = -\cos x$$
This result shows that differentiating $\cos x$ twice returns the negative of the original function.