Question

Find the partial derivatives of the function

Answer 1

\[f(x,y)=\int\limits_{x}^{y}\cos(-4t^{2}-6t+2)dt\]
find \[f_{x}(x,y) and f _{y}(x,y)\]

Answer 2

\[correction \int\limits_{y}^{x}\]

Answer 3

Hey Isabella :)

Hmm so I'm thinking we'll need to apply the Fundamental Theorem of Calculus, Part 1:\[\Large\rm \frac{d}{dx}\int\limits_c^x f(t)dt=f(x)\]

Answer 4

So like, when you're looking for the partial (with respect to x), you'll treat y as constant, just like in our Fundamental Theorem, yes?

Answer 5

\[\Large\rm f(x,y)=\int\limits\limits_{y}^{x}\cos(-4t^{2}-6t+2)dt\]So when we're taking our partial with respect to x:\[\Large\rm \frac{\partial }{\partial x}f(x,y)=\frac{\partial }{\partial x}\int\limits\limits\limits_{y}^{x}\cos(-4t^{2}-6t+2)dt\]We can think of it like this:\[\Large\rm \frac{\partial }{\partial x}f(x,y)=\frac{d}{dx}\int\limits\limits\limits_{c}^{x}\cos(-4t^{2}-6t+2)dt\]And apply our Fundamental Theorem! :)

Answer 6

Isabelleeee D:
Where hast thou gonest!