# Acceleration as the second derivative of displacement function

Let $$x$$ be displacement as a function of time $$t$$ and some other physical quantity $$k$$ such that

$$x = f(t,k)$$

Now,

1) Will the acceleration $$a$$ be $$\frac{\partial^2 x}{\partial t^2}$$ or $$\frac{d^2 x}{dt^2}$$ ?

2) Will both of the expressions yield the same result?

Thank You

• What do you exactly mean by some other physical quantity $k$? In classical mechanics, $x$ is to be treated simply as $x(t)$ because there is always a specific position of a particle at each value of time. And any other physical quantity is to be treated as $k(x(t),p(t),t)$ or $k(x(t),\dot{x}(t),t)$. – Dvij D.C. Feb 28 '19 at 18:31
• My use of $k$ was just to make $x$ an implicit function, I just wanted to know that will $a$ will be defined as the second partial derivative of $x$ wrt $t$ or second derivative of $x$ wrt $t$ .... – StaticESC Feb 28 '19 at 19:23
• What is your specific application of that? – flaudemus Feb 28 '19 at 19:34
• consider a situation where x is a function of time and some additional parameter which cannot be differentiated in terms of t, so just differentiating x wrt t will give us (dimensionally speaking) a term of acceleration and another term which will NOT have dimension of acceleration, so i just want to clarify that if we have a function of an implicit function of x, is it correct to use partial derivative ? – StaticESC Feb 28 '19 at 19:45
• Say in some problem the x coordinate depends as x = t+y² taking partial derivative wouldn't help here as we don't know time dependence of y – Aditya Garg Feb 28 '19 at 19:50