Kit Library / Mathematics / Calculus
⚡ Topic Learning KitPartial Derivatives
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What you'll study, topic by topic
Partial Derivatives
~8 min · full explanation, examples & memory tricks in the app
Try the smart MCQs from this kit
15 questions laddered from warm-up to topper-level, each with an explanation. A taste:
What does the partial derivative $\frac{\partial f}{\partial x}$ measure?
BeginnerShow answer & explanation
The rate of change of $f$ with respect to $x$, holding all other variables constant
The partial derivative with respect to $x$ measures how $f$ changes as $x$ changes, while keeping all other independent variables fixed.
If $f(x,y) = x^2 y + 3xy^3$, what is $\frac{\partial f}{\partial x}$?
IntermediateShow answer & explanation
$2xy + 3y^3$
Differentiate with respect to $x$ treating $y$ as a constant: $\frac{\partial}{\partial x}(x^2 y) = 2xy$ and $\frac{\partial}{\partial x}(3xy^3) = 3y^3$. So the result is $2xy + 3y^3$.
If $f(x,y) = e^{xy} \sin x$, what is $\frac{\partial f}{\partial y}$?
IntermediateShow answer & explanation
$x e^{xy} \sin x$
Treat $x$ as a constant. The derivative of $e^{xy}$ with respect to $y$ is $x e^{xy}$, and $\sin x$ is a constant factor. So $\frac{\partial f}{\partial y} = x e^{xy} \sin x$.
If $f(x,y,z) = x^3 y^2 z$, what is $\frac{\partial f}{\partial z}$?
BeginnerShow answer & explanation
$x^3 y^2$
Differentiate with respect to $z$, treating $x$ and $y$ as constants. The derivative of $z$ is 1, so the result is $x^3 y^2$.
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