/sci/ - Science & Math

Why can I plug in [math]y[/math] before I take the derivative? I thought you couldn't plug in a number before you take the derivative. For instance, with [math]V=\frac{1}{3}\pir^{2}h[/math], where we're looking for [math]\frac{dh}{dt}[/math] and we're given both [math]r[/math] and [math]\frac{dV}{dt}[/math] , you cannot plug in [math]r[/math] , take the derivative, plug in [math]\frac{dV}{dt}[/math] , and find [math]\frac{dh}{dt}[/math] . But in the case of what's happening in my image, it seems that you can plug in [math]y[/math] before you take the derivative, and then proceed to take the derivative and solve for [math]\frac{dx}{dt}[/math] . What's the difference here?

>>11268274
Why can I plug in [math]y[/math] before I take the derivative? I thought you couldn't plug in a number before you take the derivative. For instance, with [math]V=(\frac{1}{3})(\pi)(r^2)h[/math], where we're looking for [math]\frac{dh}{dt}[/math] and we're given both [math]r[/math] and [math]\frac{dV}{dt}[/math], you \mathbf{cannot} plug in [math]r[/math], take the derivative, plug in [math]\frac{dV}{dt}[/math], and find [math]\frac{dh}{dt}[/math]. But in the case of what's happening in my image, it seems that you \mathbf{can} plug in [math]y[/math] before you take the derivative, and then proceed to take the derivative and solve for [math]\frac{dx}{dt}[/math]. What's the difference here?