# Partial derivative

1. Aug 5, 2017

### dyn

1. The problem statement, all variables and given/known data
The question asks to calculate ∂f/∂x for f(x,y,t) = 3x2 + 2xy + y1/2t -5xt where x(t) = t3 and y(t) = 2t5

2. Relevant equations
The answer is given as ∂f/∂x = 6x + 2y - 5t

3. The attempt at a solution
I'm confused because the answer given seems to treat x,y ,t as independent variables and the answer given is just a partial derivative treating y and t as constant. But really x,y.t are all dependent on each other. Is it even possible to obtain a partial derivative with respect to x in this case ?

2. Aug 5, 2017

### andrewkirk

The information that x(t) = t3 and y(t) = 2t5 is a red herring. When taking a partial derivative of f with respect to x, we look only at the explicit, direct role of x in the formula for f(x,y,t). We ignore any dependencies, and that leads to the quoted result. If we want to take dependencies into account, we take a total derivative, which is written $\frac{df}{dx}$, rather than the partial derivative $\frac{\partial f}{\partial x}$.

The Insight article on partial derivatives gives more background on partial derivatives, and the nature of these distinctions.

3. Aug 5, 2017

### dyn

Thanks. And would I be correct that the total derivative is
df/dx = ∂f/∂t + (∂f/∂x)dx/dt + (∂f/∂y)dy/dt ?

4. Aug 6, 2017

### andrewkirk

That's the formula for the total derivative with respect to t.
The total derivative wrt x uses the same formula, but swaps the role of t and x on the RHS, giving
$$\frac{df}{dx} = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} \frac{d t}{dx} + \frac{\partial f}{\partial y} \frac{dy}{dx}$$

5. Aug 6, 2017