Partial Derivatives, Jacobians & Gradients Practice Questions, Quiz, and Step-by-Step Lesson - improve your math skills with focused questions and clear explanations.
Partial Derivatives, Jacobians & Gradients Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice multivariable differentiation: computing partial derivatives while holding other variables fixed, forming gradients, using \(D_u f=\nabla f\cdot u\) for unit directions, reading level sets, building Jacobian matrices for maps \(F:\mathbb{R}^n\to\mathbb{R}^m\), applying the chain rule, recognizing Hessians and mixed partials, writing tangent-plane linearizations, and checking when a nonzero Jacobian determinant gives local invertibility. If you want a refresher, open the lesson for short worked examples and quick checks.
How this multivariable differentiation practice works
1. Take the quiz: answer questions about partial derivatives, gradients, Jacobians, directional derivatives, and chain rules.
2. Open the lesson: review definitions, recognition tests, worked examples, and single-answer checks.
3. Retry: return to the quiz and decide which derivative object each problem is asking for.
What you will learn in the partial derivatives, Jacobians, and gradients lesson
Partial derivatives
Hold other variables fixed: \(f_x\) differentiates only the \(x\)-dependence
Mixed partials: \(f_{xy}\) and \(f_{yx}\) agree under the usual continuity hypotheses
Continuous first partials near a point are a strong enough condition for differentiability there
Gradients and directions
Gradient: \(\nabla f=(f_{x_1},\ldots,f_{x_n})\) for scalar-valued \(f\)
Directional derivative: \(D_u f(a)=\nabla f(a)\cdot u\) when \(u\) is a unit vector
The gradient is normal to regular level sets and points in the steepest-increase direction
Jacobian matrices
Rows are outputs, columns are inputs: \(J_F\) is \(m\times n\) for \(F:\mathbb{R}^n\to\mathbb{R}^m\)
For square maps, \(\det J_F\) measures local area or volume scaling and orientation
The multivariable chain rule is matrix multiplication of derivative matrices
Theorem checks and traps
Nonzero \(\det J_F(a)\) gives a local inverse for a differentiable square map with the right smoothness
A regular level set has nonzero gradient, so the gradient supplies a normal direction
Do not confuse existing partial derivatives with full differentiability or a valid tangent plane
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing multivariable differentiation.
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Multivariable Calculus & Differential Methods
Partial Derivatives, Jacobians & Gradients Lesson
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Lesson overview
Purpose: Build a reliable toolkit for multivariable differentiation: compute partial derivatives, assemble gradients and Jacobians, use directional derivatives, apply chain rules, read Hessians and mixed partials, make first-order linear approximations, and recognize when theorem hypotheses matter.
Success criteria
Compute \(f_x,f_y,\ldots\) by holding the other variables fixed.
Form \(\nabla f\) for scalar-valued functions.
Use \(D_u f(a)=\nabla f(a)\cdot u\) for unit directions.
Identify gradients as normals to regular level sets.
Build \(J_F\) with rows for outputs and columns for inputs.
Interpret \(\det J_F\) for square maps as local scaling and orientation.
Apply the chain rule in scalar and matrix form.
Use Hessians, mixed partials, and first-order linearization correctly.
Check differentiability and inverse-function hypotheses before applying theorems.
Key vocabulary
Partial derivative: one-variable derivative with all other variables held fixed.
Gradient: vector of first partial derivatives for scalar \(f\).
Directional derivative: rate of change in a specified unit direction.
Jacobian: derivative matrix of a vector-valued map.
Hessian: matrix of second partial derivatives of a scalar function.
Linearization: best first-order affine approximation near a differentiability point.
Quick pre-check
Pre-check: For a scalar function \(f(x,y)\), what is \(\nabla f\)?
Hint: A gradient collects all first partial derivatives of one scalar function.
Partial derivatives are ordinary derivatives with the other variables frozen
Learning goal: Decide which variables move, which stay fixed, and what a first partial derivative does and does not prove.
Key idea
For \(f(x,y)\), \(f_x\) means differentiate with respect to \(x\) while treating \(y\) as a constant. Likewise, \(f_y\) treats \(x\) as constant. The notation records a local rate in one coordinate direction; by itself, existence of partial derivatives does not always guarantee full differentiability.
Recognition checklist
Identify the variable of differentiation.
Treat every other variable as a constant coefficient.
Apply ordinary product, quotient, and chain rules as needed.
For products like \(g(x)h(y)\), differentiate only the factor depending on the active variable.
After differentiating, substitute the point only if the problem asks for a value.
For differentiability claims, look for stronger conditions such as continuous first partials nearby.
Worked example
Example: For \(f(x,y)=xe^y+y^2\), compute \(f_x\) and \(f_y\).
For \(f_x\), \(e^y\) and \(y^2\) are constant with respect to \(x\), so \(f_x=e^y\). For \(f_y\), \(x\) is constant, so \(f_y=xe^y+2y\).
Try it
Try it: For \(f(x,y)=x^2y+\sin x\), what is \(f_y\)?
Hint: With respect to \(y\), the term \(\sin x\) is constant.
The gradient packages every coordinate rate into one vector
Learning goal: Use gradients to compute directional derivatives and read level-set geometry.
Key idea
For \(f:\mathbb{R}^n\to\mathbb{R}\), \(\nabla f(a)\) is the vector of first partial derivatives at \(a\). If \(u\) is a unit vector, then \(D_u f(a)=\nabla f(a)\cdot u\). The largest possible directional derivative is \(\|\nabla f(a)\|\), attained in the gradient direction when the gradient is nonzero.
Gradient facts
The gradient points in the direction of steepest increase.
The negative gradient points in the direction of steepest decrease.
Directions perpendicular to \(\nabla f(a)\) have first-order change \(0\).
On a regular level set \(f=c\), the gradient is normal to the level set.
The directional derivative formula requires \(u\) to be a unit vector.
Worked example
Example: Let \(f(x,y)=x^2+y^2\). Find \(D_u f(1,2)\) for \(u=(1/\sqrt5,2/\sqrt5)\).
\(\nabla f=(2x,2y)\), so \(\nabla f(1,2)=(2,4)\). Then \(D_u f(1,2)=(2,4)\cdot(1/\sqrt5,2/\sqrt5)=10/\sqrt5=2\sqrt5\).
Try it
Try it: If \(\nabla f(a)=(3,4)\), what is the maximum directional derivative over all unit directions?
Hint: The maximum directional derivative is the length of the gradient.
A Jacobian is the derivative matrix of a vector-valued map
Learning goal: Build the Jacobian with the correct shape and interpret its determinant when the map is square.
Key idea
For \(F=(F_1,\ldots,F_m):\mathbb{R}^n\to\mathbb{R}^m\), the Jacobian matrix is \(J_F=(\partial F_i/\partial x_j)\). It has \(m\) rows and \(n\) columns: one row for each output component and one column for each input variable.
Jacobian uses
Approximate \(F(a+h)\) by \(F(a)+J_F(a)h\).
Apply the chain rule as \(J_{F\circ G}(x)=J_F(G(x))J_G(x)\).
For \(F:\mathbb{R}^n\to\mathbb{R}^n\), \(|\det J_F|\) is the local volume scale factor.
A positive determinant preserves orientation; a negative determinant reverses orientation.
A nonzero determinant is the key local-invertibility test for square maps under the standard smoothness hypotheses.
Worked example
Example: For \(F(x,y)=(x+y,xy)\), find \(J_F\) and \(\det J_F(3,1)\).
The rows come from the two components: \[J_F(x,y)=\begin{pmatrix}1&1\\ y&x\end{pmatrix}.\] Its determinant is \(x-y\), so at \((3,1)\) it equals \(2\).
Try it
Try it: If \(F:\mathbb{R}^3\to\mathbb{R}^2\), what is the size of \(J_F\)?
Hint: Count outputs for rows and inputs for columns.
The multivariable chain rule multiplies derivative data in the correct order
Learning goal: Choose the scalar or matrix chain rule that matches the dependency diagram.
Key idea
If \(z=f(x(t),y(t))\), then \(dz/dt=f_x\,dx/dt+f_y\,dy/dt\). More generally, the derivative of a composition is obtained by multiplying Jacobian matrices, with the outside derivative evaluated at the inside function.
Chain rule patterns
Path into scalar field: \(d(f(r(t)))/dt=\nabla f(r(t))\cdot r^\prime(t)\).
Map after map: \(J_{F\circ G}=J_F(G)J_G\).
Coordinate change: differentiate the new variables first, then feed them into the outer derivatives.
Check dimensions: the matrix product should have rows from the final output and columns from the original input.
Worked example
Example: Let \(f(x,y)=x^2y\), \(x=t\), and \(y=t^2\). Compute \(dz/dt\) for \(z=f(x(t),y(t))\).
Directly, \(z=t^2\cdot t^2=t^4\), so \(dz/dt=4t^3\). By the chain rule, \(f_x=2xy=2t^3\), \(x^\prime=1\), \(f_y=x^2=t^2\), and \(y^\prime=2t\), giving \(2t^3+2t^3=4t^3\).
Try it
Try it: If \(z=f(x(t),y(t))\), which formula is the chain rule?
Hint: Add the contribution through each intermediate variable.
Second partials describe how the first derivative changes
Learning goal: Recognize the Hessian matrix, mixed partial notation, and the condition that lets mixed partials commute.
Key idea
For scalar \(f\), the Hessian \(H_f\) is the matrix of second partial derivatives. Its diagonal entries are pure second partials such as \(f_{xx}\), and its off-diagonal entries are mixed partials such as \(f_{xy}\) and \(f_{yx}\). If the mixed second partials are continuous near a point, then \(f_{xy}=f_{yx}\) at that point.
Hessian facts
The Hessian of \(f(x,y)\) is \(\begin{pmatrix}f_{xx}&f_{xy}\\ f_{yx}&f_{yy}\end{pmatrix}\).
For a unit direction \(u\), the second directional derivative is \(u^T H_f(a)u\).
For a separated sum \(g(x)+h(y)\), the mixed partial \(f_{xy}\) is \(0\) whenever the derivatives exist.
Symmetry of the Hessian needs hypotheses; it is not just notation.
The Hessian is central for local shape, but the gradient gives the first-order critical point condition.
Worked example
Example: For \(f(x,y)=x^2y+y^2\), compute \(f_{xy}\) and \(f_{yx}\).
First \(f_x=2xy\), so \(f_{xy}=2x\). Also \(f_y=x^2+2y\), so \(f_{yx}=2x\). The mixed partials agree for this smooth function.
Try it
Try it: If \(f_{xy}\) and \(f_{yx}\) are continuous near a point, what follows at that point?
Hint: Continuous mixed second partials allow the order of differentiation to be switched.
Differentiability turns derivative data into a first-order model
Learning goal: Use gradients and Jacobians for local approximation, tangent planes, and the first-order critical point condition.
Key idea
If \(f\) is differentiable at \(a\), then \(f(a+h)\approx f(a)+\nabla f(a)\cdot h\). For \(z=f(x,y)\), the tangent plane at \((a,b)\) is \(z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\). At an interior local extremum where \(f\) is differentiable, the gradient must be \(0\).
Linearization patterns
For scalar \(f\), linearization uses the dot product with \(\nabla f\).
For vector-valued \(F\), linearization uses \(J_F(a)h\).
For a regular level curve \(f=c\), a tangent vector is perpendicular to \(\nabla f\).
At an interior differentiable local extremum, \(\nabla f=0\).
If \(f_x=f_y=0\), the point is critical, but classification is separate.
Do not write a tangent plane from partials unless differentiability is justified.
Worked example
Example: Suppose \(f(1,2)=5\), \(f_x(1,2)=3\), and \(f_y(1,2)=-2\). Approximate \(f(1.1,1.97)\).
Here \(h=0.1\) and \(k=-0.03\). The linear approximation gives \(5+3(0.1)-2(-0.03)=5+0.30+0.06=5.36\).
Try it
Try it: If \(f_x(a,b)=2\) and \(f_y(a,b)=-1\), what is the first-order change for an input change \((h,k)\)?
Hint: Multiply each coordinate change by the matching partial derivative and add.
The common mistakes are mostly object-type mistakes
Learning goal: Finish by separating scalar functions, vector-valued maps, gradients, Jacobians, Hessians, and determinant facts.
Common traps
Wrong frozen variable: in \(f_x\), every non-\(x\) variable is constant.
Nonunit direction: normalize the direction before using \(D_u f=\nabla f\cdot u\).
Wrong Jacobian shape: rows are outputs and columns are inputs.
Determinants only for square Jacobians: \(F:\mathbb{R}^3\to\mathbb{R}^2\) has no Jacobian determinant.
Partials are not enough: existing first partials alone need not prove differentiability.
Missing theorem hypotheses: local inverses and mixed-partial symmetry require regularity assumptions.
Worked example
Example: For \(F(x,y)=(x+y,x-y)\), what does \(\det J_F=-2\) tell you locally?
The determinant is nonzero, so the map is locally invertible. Its absolute value \(2\) is the local area scale factor, and the negative sign means the map reverses orientation.
Try it
Try it: If a square Jacobian determinant is negative at a point, the local map reverses:
Hint: The sign of a determinant records whether orientation is preserved or reversed.
Final recap
Partial derivatives are one-coordinate rates.
The gradient is the vector of first partials for scalar \(f\).
Directional derivatives use the dot product with a unit direction.
Gradients are normal to regular level sets.
The Jacobian of \(F:\mathbb{R}^n\to\mathbb{R}^m\) is \(m\times n\).
The determinant of a square Jacobian gives local scaling and orientation information.
The chain rule is dot products for scalar paths and matrix multiplication for maps.
The Hessian contains second partials; mixed partials commute under continuity hypotheses.
Linearization requires differentiability, not just formal partial derivatives.
Next step: Close this lesson and try the quiz again. For each question, first identify whether it asks for a scalar, a vector, a matrix, or a determinant.
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