Single Variable Calculus

Differentiation

01The Derivative as the Slope of the Tangent Line 02Secant Lines and the Limit Definition of the Derivative 03Differentiating 1/x Straight From the Definition 04Tangents to 1/x and the Triangle of Area 2 05Newton and Leibniz Notation, and the Power Rule Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01The Derivative as an Instantaneous Rate of Change 02Rates of Change Without Time: Gradient and Sensitivity 03Easy Limits, 0/0, and One-Sided Limits 04Continuity, Jumps, and Removable Discontinuities 05Infinite Discontinuities: 1/x and Its Derivative 06Differentiable Implies Continuous Problem set0/10 Problem set 20/10 Practice∞

01Two Kinds of Formula, and the Derivative of Sine 02The Derivative of Cosine, and Two Limits at Zero 03Bow and Bowstring: Proving the Two Trig Limits 04Why the Trig Derivatives Hold Only in Radians 05A Geometric Proof: Sine as a Height on the Circle 06Finishing the Proof, and the Product and Quotient Rules Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01Proving the Product Rule: Change One Factor at a Time 02The Quotient Rule, and the Power Rule for Negative Exponents 03The Chain Rule: Differentiating a Function of a Function 04Higher Derivatives: The Sine Cycle and Three Notations 05The nth Derivative of xⁿ Is n Factorial, by Induction Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01The Power Rule for Rational Exponents 02The Slope of a Circle, Two Ways 03Implicit Differentiation of a Quartic Curve 04Inverse Functions and the Reflection Across y = x 05Derivatives of the Arctangent and Arcsine Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Implicit Differentiation of a Quartic Curve

When the explicit branches are nested roots, how can you still get the slope of y^4 + xy^2 - 2 = 0 without ever differentiating them?

Loading…

←Previous The Slope of a Circle, Two Ways Next Inverse Functions and the Reflection Across y = x →

Your summary note

1. The explicit branches from the quadratic in $y^2$
  Apply the quadratic formula in $y^2$ to get $y^2=\frac{-x\pm\sqrt{x^2+8}}{2}$ , then show the inner minus sign gives no real $y$ and record the two real branches $y=\pm\sqrt{\frac{-x+\sqrt{x^2+8}}{2}}$ .
2. Differentiating implicitly with the chain and product rules
  Differentiate the equation term by term to $4y^3y'+y^2+x(2yy')=0$ , then factor out $y'$ and solve, writing the slope formula $y'=\frac{-y^2}{4y^3+2xy}$ .
3. Using the formula at the point $(1,1)$
  State that the formula needs both coordinates of a point, check that $(1,1)$ satisfies the equation, and substitute to reach the slope $-\frac{1}{6}$ .
4. What the method removes and what it keeps
  Record that finding $y$ at a general $x$ still demands the nested-root formula, and that the gain is a fast algebraic slope obtained without ever differentiating that formula.
Attempt 1 of 2