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Partial Derivatives Example 2
In multivariable calculus you may be asked to find the partial derivatives.
When deriving with respect to a variable, just treat all other variables as a constant.
Example problem: Find all first partial derivatives of \(f(x,t) = {e^{ – t}}\cos (\pi x)\)
We’ll find the partial derivative with respect to x first, so treat everything else as a constant.
Since x is inside the argument of a trig, we need to derive the trig, so it turns from cosine to negative sine.
We also need to multiply it by the derivative of of \(\pi \) times x, which is \(\pi \) (this is the regular chain rule for finding the derivatives of trigonometric functions).
\[\frac{{\partial f}}{{\partial x}} = – \pi {e^{ – t}}\sin (\pi x)\]
Now find the derivative with respect to t, so you treat x as a constant.
Now you don’t touch the sine at all because there’s no t in the arguments of the sine function.
The derivative of e to the negative t is the derivative of -t, which is -1.
\[\frac{{\partial f}}{{\partial y}} = – {e^{ – y}}cos (\pi x)\]