Q.If $y = \sin(\sin x)$, prove that $\dfrac{d^2 y}{dx^2} + \tan x\, \dfrac{dy}{dx} + y\cos^2 x = 0$.

Successive Differentiation — Repeated Slopes

Differentiate a function, then differentiate the result, then differentiate that, and so on. Each pass produces a new function describing a deeper layer of change. The physics picture makes it concrete: position $\to$ velocity (1st derivative) $\to$ acceleration (2nd) $\to$ jerk (3rd). Every step asks the same question: "how does the previous rate of change itself change?"

The definition and notation

If $y=f(x)$, its successive derivatives are written $y_1,y_2,\dots,y_n$, equivalently $f'(x),f''(x),\dots,f^{(n)}(x)$ or $\dfrac{dy}{dx},\dfrac{d^2y}{dx^2},\dots,\dfrac{d^ny}{dx^n}$. Each one is the derivative of the previous:

Seeing the pattern

Take $y=x^4$: $\;y_1=4x^3,\;y_2=12x^2,\;y_3=24x,\;y_4=24,\;y_5=0$. Each differentiation drops the degree by one, so a degree-$n$ polynomial has a constant $n$th derivative and vanishing higher ones. Other families behave differently: $e^{ax}$ never dies out ($\frac{d^n}{dx^n}e^{ax}=a^n e^{ax}$), and $\sin x$ cycles every four steps ($\sin\to\cos\to-\sin\to-\cos$).