Implicit Differentiation
- Page ID
- 624
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Implicit and Explicit Functions
An explicit function is an function expressed as y = f(x) such as
\[ y = \text{sin}\; x \nonumber \]
y is defined implicitly if both x and y occur on the same side of the equation such as
\[ x^2 + y^2 = 4 \nonumber \]
we can think of y as function of x and write:
\[ x^2 + y(x)^2 = 4 \nonumber \]
Implicit Differentiation
To find dy/dx, we proceed as follows:
- Take d/dx of both sides of the equation remembering to multiply by y' each time you see a y term.
- Solve for y'
Example
Find dy/dx implicitly for the circle
\[ x^2 + y^2 = 4 \nonumber \]
Solution
-
d/dx (x2 + y2) = d/dx (4)
or
2x + 2yy' = 0
-
Solving for y, we get
2yy' = -2x
y' = -2x/2y
y' = -x/y
Example:
Find y' at (4,2) if
\[ xy + \dfrac{x}{y} = 10 \nonumber \]
Solution:
-
\[ (xy)' + \left(\dfrac{x}{y}\right)' = (5)' \nonumber \]
Using the product rule and the quotient rule we have -
\[ xy' + y + \dfrac{y - xy'}{ y^2} = 0 \nonumber \]
-
Now plugging in x = 4 and y = 2,
2 - 4y'
4y' + 2 +
2216y' + 8 + 2 - 4y' = 0 Multiply both sides by 4
12y' + 10 = 0
12y' = -10
y' = -5/6
Exercises
-
Let
\[ 3x^2 - y^3 = 4x \text{cos}\; x + y^2 \nonumber \]
Find dy/dx -
Find dy/dx at (-1,1) if
\[ (x + y)^3 = x^3 + y^3 \nonumber \] -
Find dy/dx if
\[ x^2 + 3xy + y^2 = 1 \nonumber \] -
Find y'' if
\[ x^2 - y^2 = 4 \nonumber \]