Lesson 14 Implicit Differentiation

14.1 Lecture Content

14.2 Lecture Notes

14.2.1 Higher-Order Derivatives

Higher-order derivatives are the successive derivatives of a function.

First derivative: \(f'(x)\) — represents the rate of change or slope.
Second derivative: \(f''(x)\) — measures the rate of change of the first derivative (i.e., concavity or acceleration).
Third and beyond: \(f^{(3)}(x), f^{(4)}(x), \ldots\) are called third, fourth, etc., derivatives.

Notation:

\(f^{(n)}(x)\) is the \(n\)th derivative.
In physics, these can represent velocity, acceleration, jerk, etc.

Example: If \(f(x) = x^3\):

\(f'(x) = 3x^2\)
\(f''(x) = 6x\)
\(f^{(3)}(x) = 6\)

14.2.2 Implicit Differentiation

Implicit differentiation is used when a function is not given in the form \(y = f(x)\), but instead in an implicit form, like an equation involving both \(x\) and \(y\).

14.2.2.1 Steps:

Differentiate both sides of the equation with respect to \(x\).
- Treat \(y\) as a function of \(x\) (i.e., apply the chain rule to any term involving \(y\)).
Apply the chain rule:
- When differentiating terms with \(y\), include \(\frac{dy}{dx}\) (or \(y'\)).
Solve for \(\frac{dy}{dx}\) (i.e., isolate the derivative on one side).

Example: Given \(x^2 + y^2 = 25\), find \(\frac{dy}{dx}\):

Differentiate both sides: \[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25) \Rightarrow 2x + 2y \frac{dy}{dx} = 0 \]
Solve for \(\frac{dy}{dx}\): \[ 2y \frac{dy}{dx} = -2x \Rightarrow \frac{dy}{dx} = -\frac{x}{y} \]

14.3 Examples

Video: Solutions

Use implicit differentiation to determine \(\frac{dy}{dx}\) given the equation \(x^5 + y^6 = -7\).
Given \(4x^2 + 2x + xy = 2\), and \(y(2) = -9\), find \(y'(2)\) by implicit differentiation.
Find the slope of the tangent line to the curve \(4x^2 + 2xy - 4y^3 = 6\) at the point \((1, -1)\).

14.4 Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

Find the first \(n\) derivatives of \[f(x) = -3x^3 + 4x.\]
Find \(dy/dx\) given \[x^2 y - y^2 = 4.\]
Find \(dy/dx\) given \[3xy - \frac{y}{x} = 2.\]
Find \(dy/dx\) given \[y \ln x + y = 2.\]
Find \(dy/dx\) given \[x e^y = y e^x.\]
Differentiate \[y = \frac{(x+1)(x+2)(x+3)(x+4)}{x(x-1)(x-2)(x-3)}.\]
Find the tangent line to \[e^{-xy} + 2x = 1\] at \(x = -1\).
(Applied) An employment research company estimates that the value of a recent MBA graduate to an accounting company is \[ V = 3e^2 + 5g^3, \] where \(V\) is the value of the graduate, \(e\) is the number of years of prior business experience, and \(g\) is the graduate school grade point average. If \(V\) is fixed at 200, find \(de/dg\) when \(g = 3.0\) and interpret the result.
(Applied) Use logarithmic differentiation to give a proof of the quotient rule.

14.5 Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

Find the first \(n\) derivatives of \[f(x) = (-2x + 1)^3.\]
Find \(dy/dx\) given \[\ln(y^2 - y) + x = y.\]
Find \(dy/dx\) given \[e^x y = 1.\]
Differentiate \[y = \frac{(3x + 1)^2}{4x(2x - 1)^3}.\]
Find the tangent line to \[4x^2 + 2y^2 = 12\] at \((1, -2)\).

14.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill	D	CON	COM
Find higher order derivatives of a function.
Use implicit differentiation to find the derivative.
Use logarithmic differentiation to find the derivative.