Multivariable calculus

Partial derivatives

Differentiate with respect to one variable and hold the rest still.

• Partial Derivatives of x^2 y^3: Method and AnswersBoth partials of x^2 y^3: f_x = 2xy^3 and f_y = 3x^2 y^2. See the hold-the-other-variable-fixed method and the product rule slip to avoid.
• Partial Derivatives of x^3 + y^3 - 3xyFor x^3 + y^3 - 3xy the partials are f_x = 3x^2 - 3y and f_y = 3y^2 - 3x. Work term by term, then use them to find the critical points.
• Partial Derivatives of 2x^2 - 3xy + 5y^2The partials of the quadratic form 2x^2 - 3xy + 5y^2 are f_x = 4x - 3y and f_y = -3x + 10y, both linear, with a constant Hessian matrix.
• Partial Derivatives of x^3 - 3xy^2 (Monkey Saddle)For the monkey saddle x^3 - 3xy^2 the partials are f_x = 3x^2 - 3y^2 and f_y = -6xy. The second partials cancel, so the function is harmonic.
• Partial Derivatives of (x^2 y + 1)^3For (x^2 y + 1)^3 the partials are f_x = 6xy(x^2 y + 1)^2 and f_y = 3x^2 (x^2 y + 1)^2. The chain rule outer factor is shared by both.
• Partial Derivatives of (x + y)/(x^2 + y^2 + 1)For (x + y)/(x^2 + y^2 + 1) the partials are 1 - x^2 + y^2 - 2xy and 1 + x^2 - y^2 - 2xy, each over the denominator squared.
• Partials of (x^2 - y^2)/(1 + x^2 y^2)The partials of (x^2 - y^2)/(1 + x^2 y^2) simplify beautifully: 2x(1 + y^4) and -2y(1 + x^4), each over the squared denominator.
• Partial Derivatives of x/(y^2 + 1)For x/(y^2 + 1) the partials are f_x = 1/(y^2 + 1) and f_y = -2xy/(y^2 + 1)^2. No quotient rule is needed for the x partial.
• Partial Derivatives of xy^2/(1 + x^2)For xy^2/(1 + x^2) the partials are f_x = y^2(1 - x^2)/(1 + x^2)^2 and f_y = 2xy/(1 + x^2). Only the x partial needs the quotient rule.
• Partial Derivatives of x^3 y^2 - 2xy^4For x^3 y^2 - 2xy^4 the partials are f_x = 3x^2 y^2 - 2y^4 and f_y = 2x^3 y - 8xy^3, with mixed second partials that agree.
• Partial Derivatives of 5x^2 y - 3x + 4y - 7For 5x^2 y - 3x + 4y - 7 the partials are f_x = 10xy - 3 and f_y = 5x^2 + 4. Constants and single-variable terms drop out in the right places.
• Partial Derivatives of x^4 + y^4 - 4xyFor x^4 + y^4 - 4xy the partials are f_x = 4x^3 - 4y and f_y = 4y^3 - 4x, which vanish at (0, 0), (1, 1), and (-1, -1).
• Partial Derivatives of sin(x)cos(y)Both partial derivatives of f(x,y) = sin(x)cos(y), worked one variable at a time, with the standing wave identity that links them.
• Partial Derivatives of sin(xy)Both partials of f(x,y) = sin(xy) by the chain rule, why the inside derivative is y one way and x the other, with a numeric check.
• Partial Derivatives of x^2 sin(y)Find both partials of f(x,y) = x^2 sin(y), a clean example of holding one variable constant, with a worked value and a common slip.
• Partial Derivatives of cos(x^2 + y^2)Both partials of the radial ripple f(x,y) = cos(x^2 + y^2), the chain rule step that produces the 2x and 2y factors, and what the gradient means.
• Partial Derivatives of e^x sin(y)Both partials of f(x,y) = e^x sin(y), the harmonic function behind complex analysis, plus a check that its Laplacian is zero.
• Partial Derivatives of arctan(xy)Both partials of f(x,y) = arctan(xy) using the inverse tangent derivative and the chain rule, with the bounded behaviour explained.
• Partial Derivatives of sin(x + 2y)Both partials of the travelling wave f(x,y) = sin(x + 2y), where the coefficient 2 lands, and the wave equation the function satisfies.
• Partial Derivatives of sin^2(x) + cos^2(y)Both partials of f(x,y) = sin^2(x) + cos^2(y), why the Pythagorean identity does not apply, and the double angle form of the answers.
• Partials of sin(x) / (2 + cos(y))Both partials of f(x,y) = sin(x)/(2 + cos(y)), one by constant multiple and one by the quotient rule, with the sign carefully tracked.
• Partials of x sin(y) + y cos(x)Both partials of f(x,y) = x sin(y) + y cos(x), where every term contributes to both answers, plus the tangent plane at the origin.
• Partials of ln(3 + sin(x)cos(y))Both partials of f(x,y) = ln(3 + sin(x)cos(y)) by the logarithm chain rule, with the shared denominator explained and the range checked.
• Partial Derivatives of y^2 cos(xy)Both partials of f(x,y) = y^2 cos(xy), where one needs only the chain rule and the other needs the product rule as well.
• Partial Derivatives of e^(xy): Method and MistakesThe partials of e^(xy) are y e^(xy) in x and x e^(xy) in y. See the chain rule step, the two wrong answers, and a check at a point.
• Partial Derivatives of e^(-x^2-y^2)The partials of the Gaussian bump e^(-x^2-y^2) are -2x e^(-x^2-y^2) and -2y e^(-x^2-y^2). Chain rule step, sign check, and where the slope is steepest.
• Partial Derivatives of ln(x^2 + y^2 + 1)For ln(x^2 + y^2 + 1) the partials are 2x/(x^2+y^2+1) and 2y/(x^2+y^2+1). The u prime over u form, the domain, and the log-of-a-sum trap.
• Partial Derivatives of ln(e^x + e^y)The partials of ln(e^x + e^y) are e^x/(e^x + e^y) and e^y/(e^x + e^y). They are positive, they add to exactly 1, and the function has no critical point.
• Partial Derivatives of e^x ln(1 + y^2)For e^x ln(1 + y^2) the partials are e^x ln(1 + y^2) and 2y e^x/(1 + y^2). A separated product where each partial touches only one factor.
• Partial Derivatives of x^2 e^(-y)For x^2 e^(-y) the partials are 2x e^(-y) and -x^2 e^(-y). One variable per factor, with the minus sign coming from the exponent.
• Partial Derivatives of e^(x^2 - y^2)For e^(x^2 - y^2) the partials are 2x e^(x^2-y^2) and -2y e^(x^2-y^2). The chain rule on a saddle exponent, and the sign students drop.
• Partial Derivatives of ln(1 + e^(xy))For ln(1 + e^(xy)) the partials are y e^(xy)/(1 + e^(xy)) and x e^(xy)/(1 + e^(xy)). Two stacked chain rules, and the logistic factor they produce.
• Partial Derivatives of e^x/(1 + e^y)For e^x/(1 + e^y) the partials are e^x/(1 + e^y) and -e^x e^y/(1 + e^y)^2. Quotient rule in y only, since x never touches the denominator.
• Partial Derivatives of e^(x/(1+y^2))For e^(x/(1+y^2)) the partials are e^(x/(1+y^2))/(1+y^2) and -2xy e^(x/(1+y^2))/(1+y^2)^2. A chain rule nested inside a chain rule.
• Partial Derivatives of x e^y + y e^xFor x e^y + y e^x the partials are e^y + y e^x and x e^y + e^x. A symmetric sum where each term is handled with a constant multiple.
• Partial Derivatives of e^(xy) ln(1 + z^2)A three-variable partial derivative example: the partials of e^(xy) ln(1+z^2) are y e^(xy) ln(1+z^2), x e^(xy) ln(1+z^2), and 2z e^(xy)/(1+z^2).
• Partial Derivatives of x^2 y e^xPartial derivatives of x^2 y e^x. The x-partial needs the product rule and factors to xy(x+2)e^x; the y-partial is just x^2 e^x.
• Partial Derivatives of x sin(xy)Partial derivatives of x sin(xy). The x-partial needs the product and chain rules together: sin(xy) + xy cos(xy). The y-partial is x^2 cos(xy).
• Partial Derivatives of x ln(x^2 + y^2 + 1)Partial derivatives of x ln(x^2 + y^2 + 1). The x-partial needs the product rule and gives ln(x^2+y^2+1) + 2x^2/(x^2+y^2+1).
• Partial Derivatives of e^(xy) cos yPartial derivatives of e^(xy) cos y. The y-partial needs the product rule and gives e^(xy)(x cos y - sin y); the x-partial is y e^(xy) cos y.
• Partial Derivatives of x^2 y/(x^2 + y^2 + 1)Partial derivatives of x^2 y/(x^2 + y^2 + 1) by the quotient rule. The x-partial is 2xy(y^2 + 1) and the y-partial is x^2(x^2 - y^2 + 1), each over D squared.
• Partial Derivatives of (x^2 - y^2)/(x^2 + y^2 + 1)Partial derivatives of (x^2 - y^2)/(x^2 + y^2 + 1) by the quotient rule, with the numerators collapsing to 2x(2y^2 + 1) and -2y(2x^2 + 1).
• Partial Derivatives of sin(x)/(1 + y^2)Partial derivatives of sin(x)/(1 + y^2). The x-partial is cos(x)/(1 + y^2); the y-partial needs the quotient rule and gives -2y sin(x)/(1 + y^2)^2.
• Partial Derivatives of (x + y)/(1 + e^x)Partial derivatives of (x + y)/(1 + e^x). The x-partial uses the quotient rule with an exponential denominator; the y-partial is simply 1/(1 + e^x).
• Partial Derivatives of x arctan(xy)Partial derivatives of x arctan(xy). The x-partial combines the product and chain rules: arctan(xy) + xy/(1 + x^2 y^2). The y-partial is x^2/(1 + x^2 y^2).
• Partial Derivatives of xyz e^zPartial derivatives of f(x,y,z) = xyz e^z. Only the z-partial needs the product rule, giving xy(z + 1)e^z. The x and y partials are yz e^z and xz e^z.
• Partial Derivatives of xyz/(1 + x^2 y^2 z^2)Partial derivatives of xyz/(1 + x^2 y^2 z^2). The function depends only on the product xyz, so all three partials share the factor 1 - x^2 y^2 z^2.
• Partial Derivatives of xy e^(-x^2 - y^2)Partial derivatives of xy e^(-x^2 - y^2) by the product rule: y(1 - 2x^2)e^(-x^2-y^2) and x(1 - 2y^2)e^(-x^2-y^2), plus the five critical points.
• Partial Derivatives of sin(x^2 + y^2): Chain RuleBoth partials of sin(x^2 + y^2) come from one chain rule step: the outer cosine times the partial of the inner sum. Worked out with a numeric check.
• Partial Derivatives of e^(x y^2)The partials of e^(x y^2) are y^2 and 2xy times the function itself. The exponential is reproduced, so only the inner partial of x y^2 changes.
• Partial Derivatives of ln(1 + x^2 + y^4)The partials of ln(1 + x^2 + y^4) are 2x and 4y^3 over 1 + x^2 + y^4. One chain rule step, and the two inner partials are not parallel.
• Partial Derivatives of e^(-x^2 - 2y^2)The partials of e^(-x^2 - 2y^2) are -2x and -4y times the function. The mismatched coefficients tilt the gradient off the radial direction.
• Partial Derivatives of arctan(x/(1 + y^2))The partials of arctan(x/(1 + y^2)) are 1 + y^2 and -2xy over (1 + y^2)^2 + x^2. Clearing the compound fraction early is where the work is.
• Partial Derivatives of cos(x - y^2)The partials of cos(x - y^2) are -sin(x - y^2) and 2y sin(x - y^2). The y partial picks up two minus signs that cancel, which is where errors creep in.
• Partial Derivatives of (x^2 + y^2 + 1)^(3/2)The partials of (x^2 + y^2 + 1)^(3/2) simplify to 3x and 3y times the square root of x^2 + y^2 + 1. Power rule outside, chain rule inside.
• Partial Derivatives of cos(x^2 y): Chain RuleThe partials of cos(x^2 y) are -2xy sin(x^2 y) and -x^2 sin(x^2 y). The inner product needs the constant-multiple rule, not the product rule.
• Partial Derivatives of e^(sin x + cos y)The partials of e^(sin x + cos y) are cos x and -sin y times the function itself. A two layer composition where each variable sits in its own inner term.
• Partial Derivatives of 1/(1 + x^2 + y^2)The partials of 1/(1 + x^2 + y^2) are -2x and -2y over the square of 1 + x^2 + y^2. Rewrite as a negative power and the chain rule does the rest.
• Partial Derivatives of sqrt(1 + x^2 + 2y^2 + 3z^2)Each partial of sqrt(1 + x^2 + 2y^2 + 3z^2) is a weighted variable over the root itself: x, 2y and 3z on top, not three bare variables.
• Partial Derivatives of sin(xyz): Chain RuleThe partials of sin(xyz) are yz, xz and xy times cos(xyz). Each one keeps the product of the two variables you did not differentiate against.
• Partial Derivatives of x^2y + y^2z + z^2xAll three partial derivatives of f(x,y,z) = x^2y + y^2z + z^2x, term by term, plus a homogeneity check that catches a wrong coefficient.
• Partial Derivatives of e^(xyz)The three partials of f(x,y,z) = e^(xyz) all share the same exponential factor and differ only in the inside derivative: yz, xz or xy.
• Partial Derivatives of ln(x^2 + y^2 + z^2 + 1)Partial derivatives of f(x,y,z) = ln(x^2 + y^2 + z^2 + 1), a log chain rule where all three answers share one denominator.
• Partial Derivatives of sin(xy)cos(z)Partial derivatives of f(x,y,z) = sin(xy)cos(z), where two variables sit inside a chain rule and the third is a clean separate factor.
• Partial Derivatives of sqrt(x^2 + y^2 + z^2 + 1)Partial derivatives of the smoothed radial function f = sqrt(x^2 + y^2 + z^2 + 1), with the half power chain rule and a self checking identity.
• Partial Derivatives of (x + y)/(1 + z^2)Partial derivatives of f(x,y,z) = (x + y)/(1 + z^2), showing when the quotient rule is needed and when a constant factor does the job.
• Partial Derivatives of arctan(x + yz)Partial derivatives of f(x,y,z) = arctan(x + yz), where every answer shares one denominator and differs only in the inside derivative.
• Partial Derivatives of e^(-x^2 - y^2 - z^2)Partial derivatives of the three dimensional Gaussian f = e^(-x^2 - y^2 - z^2), with the chain rule and what the gradient says about the shape.
• Partial Derivatives of xy ln(z^2 + 1)Partial derivatives of f(x,y,z) = xy ln(z^2 + 1), where two answers need no chain rule at all and the third needs one.
• Partial Derivatives of xyz/(1 + x^2 + y^2 + z^2)The full quotient rule in three variables for f = xyz/(1 + x^2 + y^2 + z^2), with the sign pattern that makes the three answers easy to check.