2.7.1: Resources and Key Concepts

Resources

Prerequisite Topics

The following is a list of prerequisite skills (all of which can be reviewed in CRC's Corequisite Codex) needed for this section that have not already been mentioned in any previous section. If you are enrolled in a course with a Support section, some (but definitely not all) of these topics might be covered or reviewed in the Support section of your course.

• Solving Equations
  • Solving Literal Equations: After differentiating implicitly, one needs to algebraically solve for \(\frac{dy}{dx}\) (or \(y'\)), which is a literal equation solving skill.

Videos

• Implicit differentiation
  • Implicit Differentiation - Differentiating Implicitly
  • Synthesis Questions
• Derivative of inverse trigonometric functions
  • Derivatives of Inverse Trigonometric Functions: Theorem, Proof, and Example
• Derivative of inverse hyperbolic functions
  • Derivatives of Inverse Hyperbolic Functions: Theorem, Proof, and Example

Key Concepts

Definitions

• Explicit Function: A function where the dependent variable \(y\) is expressed directly in terms of the independent variable \(x\) (e.g., \(y = x^2+1\)).
• Implicit Function: A function where the relationship between \(x\) and \(y\) is defined by an equation where \(y\) is not expressed entirely in terms of \(x\) (e.g., \(y-x^2=1\) or \(x^2+y^2=25\)). An equation defines a function implicitly if the function satisfies that equation.
• Implicit Differentiation: A technique to find the derivative \(\frac{dy}{dx}\) for a function defined implicitly, by differentiating both sides of the equation with respect to \(x\) and then algebraically solving for \(\frac{dy}{dx}\).

Theorems

• Theorem: Derivatives of the Inverse Trigonometric Functions:
  • \(\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}\)
  • \(\frac{d}{dx}(\cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}\)
  • \(\frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}\)
  • \(\frac{d}{dx}(\csc^{-1}x) = -\frac{1}{x\sqrt{x^2-1}}\) (Note: text uses specific range for \(\csc^{-1}x\))
  • \(\frac{d}{dx}(\sec^{-1}x) = \frac{1}{x\sqrt{x^2-1}}\) (Note: text uses specific range for \(\sec^{-1}x\))
  • \(\frac{d}{dx}(\cot^{-1}x) = -\frac{1}{1+x^2}\)
• Derivatives of the Inverse Hyperbolic Functions:
  • \(\frac{d}{dx}(\text{sinh}^{-1}x) = \frac{1}{\sqrt{1+x^2}}\)
  • \(\frac{d}{dx}(\text{cosh}^{-1}x) = \frac{1}{\sqrt{x^2-1}}\) for \(x>1\)
  • \(\frac{d}{dx}(\text{tanh}^{-1}x) = \frac{1}{1-x^2}\) for \(|x|<1\)
  • \(\frac{d}{dx}(\text{coth}^{-1}x) = \frac{1}{1-x^2}\) for \(|x|>1\)
  • \(\frac{d}{dx}(\text{sech}^{-1}x) = -\frac{1}{x\sqrt{1-x^2}}\) for \(0 < x < 1\)
  • \(\frac{d}{dx}(\text{csch}^{-1}x) = -\frac{1}{|x|\sqrt{1+x^2}}\) for \(x \neq 0\)

Common Mistakes

• Forgetting \(\frac{dy}{dx}\) (or \(y'\)): When differentiating a term involving \(y\) with respect to \(x\), a common error is to forget to multiply by \(\frac{dy}{dx}\) as required by the Chain Rule. For example, incorrectly differentiating \(y^2\) as \(2y\) instead of \(2y\frac{dy}{dx}\).
• Algebraic Errors when Solving for \(\frac{dy}{dx}\): After differentiating both sides, errors can occur when isolating the \(\frac{dy}{dx}\) term.
• Solving for \(y\) Prematurely: While sometimes possible, trying to solve the original equation for \(y\) explicitly before differentiating can be much harder or impossible, defeating the purpose of implicit differentiation.
• Not Substituting Point Values Before Solving for \(\frac{dy}{dx}\) (When a specific point is given): When asked for the slope at a specific point, it's often easier to substitute the \(x\) and \(y\) coordinates of the point into the differentiated equation before solving for \(\frac{dy}{dx}\).
• Forgetting the Chain Rule: When differentiating functions like \(\ln(u(x))\), \(\sin^{-1}(u(x))\), or \(\text{sinh}^{-1}(u(x))\), it's crucial to multiply by \(u'(x)\).
• Ignoring Domain Restrictions: Derivatives of inverse functions (trigonometric and hyperbolic) often have specific domains where they are valid, related to the ranges of the original functions and the domains of the inverse functions. For example, \(\frac{d}{dx}(\text{cosh}^{-1}x)\) is for \(x>1\).
• Inverse Secant and Cosecant Range Ambiguity: Different textbooks may use different range restrictions for \(\sec^{-1}x\) and \(\csc^{-1}x\), which can affect the sign of their derivatives or the form involving \(|x|\). This textbook explicitly states its chosen ranges.
• Confusing Derivatives of Similar Functions: For example, mixing up the derivatives of \(\tan^{-1}x\) and \(\tanh^{-1}x\). Note that \(\frac{d}{dx}(\tanh^{-1}x) = \frac{d}{dx}(\coth^{-1}x) = \frac{1}{1-x^2}\), but their domains differ.