Implicit Differentiation Examples

• Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined in terms of one variable.

Examples of Implicit Differentiation

• Example 1: Finding $\frac{dy}{dx}$ for $x^2 + y^2 = 25$

  • Differentiate both sides with respect to $x$: $\frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)$
  • Apply the chain rule: $2x + 2y \frac{dy}{dx} = 0$
  • Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$
  • The equation $x^2 + y^2 = 25$ defines a circle of radius 5.
  • The two explicit forms of the function are $y = \pm \sqrt{25 - x^2}$.
  • Differentiating $y = \sqrt{25 - x^2}$ yields $\frac{dy}{dx} = \frac{-x}{\sqrt{25 - x^2}} = -\frac{x}{y}$.
  • Differentiating $y = -\sqrt{25 - x^2}$ yields $\frac{dy}{dx} = \frac{x}{\sqrt{25 - x^2}} = -\frac{x}{y}$.
  • At points $(-5, 0)$ and $(5, 0)$, $\frac{dy}{dx}$ is undefined.
  • At point $(3, 4)$, $\frac{dy}{dx} = -\frac{3}{4}$.
  • At point $(3, -4)$, $\frac{dy}{dx} = \frac{3}{4}$.

• Example 2: Finding the tangent line to the ellipse $x^2 - xy + y^2 = 3$ at $(-1, 1)$

  • Differentiate both sides with respect to $x$: $\frac{d}{dx}(x^2 - xy + y^2) = \frac{d}{dx}(3)$
  • Apply the product and chain rules: $2x - (y + x\frac{dy}{dx}) + 2y\frac{dy}{dx} = 0$
  • Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{y - 2x}{2y - x}$
  • At point $(-1, 1)$, $\frac{dy}{dx} = \frac{1 - 2(-1)}{2(1) - (-1)} = 1$
  • The equation of the tangent line at $(-1, 1)$ is $y - 1 = 1(x + 1)$, which simplifies to $y = x + 2$.

• Example 3: Finding $\frac{d^2y}{dx^2}$ for $x^4 + y^4 = 16$

  • Differentiate both sides with respect to $x$: $\frac{d}{dx}(x^4 + y^4) = \frac{d}{dx}(16)$
  • Apply the chain rule: $4x^3 + 4y^3 \frac{dy}{dx} = 0$
  • Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x^3}{y^3}$
  • Differentiate $\frac{dy}{dx}$ with respect to $x$ to find $\frac{d^2y}{dx^2}$:
    • $\frac{d^2y}{dx^2} = \frac{d}{dx}(-\frac{x^3}{y^3}) = -\frac{3x^2y^3 - x^3(3y^2\frac{dy}{dx})}{y^6}$
  • Substitute $\frac{dy}{dx} = -\frac{x^3}{y^3}$ into the equation:
    • $= -\frac{3x^2y^3 - 3x^3y^2(-\frac{x^3}{y^3})}{y^6}$
  • Simplify: $= -\frac{3x^2y^3 + 3\frac{x^6}{y}}{y^6} = -\frac{3x^2y^4 + 3x^6}{y^7} = -\frac{3x^2(y^4 + x^4)}{y^7}$
  • Since $x^4 + y^4 = 16$, substitute 16 into the equation:
    • $= -\frac{3x^2 \cdot 16}{y^7} = -\frac{48x^2}{y^7}$
  • Thus, $\frac{d^2y}{dx^2} = -\frac{48x^2}{y^7}$.

• Each historical text is based on a landmark Supreme Court case.

• Use these resources to determine if the Bill of Rights truly protects individual liberties.

• Write violation or no violation, the number of the amendment, and appropriate phrases from the amendment to relate to the situation.

Dred Scott v. Sandford, 1857

• Dred Scott was a slave in Missouri.
• From 1833 to 1843, Scott resided in Illinois (a free state) and the Louisiana Territory, where slavery was forbidden by the Missouri Compromise of 1820.
• Scott filed suit in Missouri court for his freedom, claiming residence in free territory made him a free man.
• After losing, Scott brought a new suit in federal court.
• Scott's master maintained that no "negro" or descendant of slaves could be a citizen per Article III of the Constitution.
• The majority held that "a negro, whose ancestors were imported into [the U.S.], and sold as slaves," whether enslaved or free, could not be an American citizen and therefore did not have standing to sue in federal court.
• Taney dismissed the case on procedural grounds because the Court lacked jurisdiction.
• Taney held that the Missouri Compromise of 1820 was unconstitutional and foreclosed Congress from freeing slaves within Federal territories.
• The opinion showed deference to Missouri courts, which held that moving to a free state did not render Scott emancipated.
• Slaves were property under the Fifth Amendment, and any law depriving a slave owner of that property was unconstitutional.
• Benjamin Robbins Curtis dissented, finding the Court lacked jurisdiction and it was unnecessary to resolve the case, and categorically opposed anti-slavery laws.
• John McLean echoed Curtis, finding descent should have been litigated, and that people of African descent could be citizens.