# The way of writing the second derivatives of $y$ with respect to $x$, in Leibniz notation [duplicate]

The way of writing the second (or higher) derivative of $$y$$ with respect to $$x$$, in the Leibniz notation is $$\dfrac{d^2y}{dx^2}$$.

Why $$d$$ on face takes number $$2$$ as its power, but in denominator $$d$$ does not take the number ($$x$$ takes $$2$$)?

They say it comes from the following formal manipulation of symbols:$$\frac{d\left (\frac{dy}{dx}\right )}{dy}=\left (\frac{d}{dx}\right )^2y=\frac{d^2y}{dx^2}.$$But that doesn't explain, exactly, why the $$d$$ in denominator doesn't take $$2$$.

• See this post and this one May 20, 2022 at 12:47
• Basically, it is a notational issue with historical and theoretical aspects: we want make a difference with the "square of..." May 20, 2022 at 13:06

Recall that the definition of the derivative is:

$$f'(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

And the second derivative is simply the derivative of the first.

$$f''(x) = \lim_{\Delta x \rightarrow 0} \frac{f'(x + \Delta x) - f'(x)}{\Delta x}$$

Plugging in the difference quotient for $$f'$$ itself, we get:

$$f''(x) = \lim_{\Delta x \rightarrow 0} \frac{\frac{f(x + 2\Delta x) - f(x + \Delta x)}{\Delta x} - \frac{f(x + \Delta x) - f(x)}{\Delta x}}{\Delta x}$$

or, simplifying a little,

$$f''(x) = \lim_{\Delta x \rightarrow 0} \frac{f(x + 2\Delta x) - 2f(x + \Delta x) + f(x)}{({\Delta x})^2}$$

The $$dx^2$$ in the denonimator directly corresponds to the $$({\Delta x})^2$$ in the denominator of this definition of $$f''$$: It actually is the square of something.

But the numerator is not $$({\Delta y})^2$$, which would be $$(f(x + \Delta x) - f(x))^2 = f(x + \Delta x)^2 - 2 f(x) f(x + \Delta x) + f(x)^2$$. So we write $$d^2y$$ instead of $$dx^2$$ as an indication that we're performing a difference operation twice, rather than squaring a difference like we do with the denominator.

• Ther formula for $f''(x)$ after "Plugging in the difference quotient for $f'$ itself" is okay for motivational purposes, but requires a proof. Jun 3, 2022 at 10:07