Difference between revisions of "Implicit Differentiation"
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We want to find derivatives of these functions without having to solve | We want to find derivatives of these functions without having to solve | ||
for <math>y</math> explicitly. We can do this by implicit differentiation, | for <math>y</math> explicitly. We can do this by implicit differentiation, | ||
| − | + | in which we take the derivative of both sides of our equation with respect | |
to <math>x</math>, and do some algebra steps to solve for <math>y'</math> (or <math>\dfrac{dy}{dx}</math> | to <math>x</math>, and do some algebra steps to solve for <math>y'</math> (or <math>\dfrac{dy}{dx}</math> | ||
| − | if you prefer), keeping in mind that <math>y</math> is a function of <math>x</math> | + | if you prefer), keeping in mind that <math>y</math> is a function of <math>x</math> throughout |
the equation. | the equation. | ||
| + | |||
| + | |||
| + | == Example 1 == | ||
| + | |||
| + | 1. Find <math>y'</math> if <math>\sin y-3x^{2}y=8</math>. | ||
| + | |||
| + | [Momentarily think of <math>y=f\left(x\right)</math> and view the equation as <math>\sin\left(f\left(x\right)\right)-3x^{2}\cdot f\left(x\right)=8</math> | ||
| + | to realize that the <math>\sin\left(f\left(x\right)\right)</math> term requires | ||
| + | the chain rule and the <math>3x^{2}\cdot f\left(x\right)</math> term needs the | ||
| + | product rule when we differentiate with respect to <math>x</math>, while the derivative | ||
| + | of 8 is just 0.] | ||
| + | |||
| + | So we get | ||
| + | \[ | ||
| + | \begin{array}{rcl} | ||
| + | \sin y-3x^{2}y & = & 8\\ | ||
| + | \left(\cos y\right)\cdot y'-\left(3x^{2}y'+6xy\right) & = & 0\quad\textrm{(derivative of both sides with respect to \ensuremath{x})}\\ | ||
| + | \left(\cos y\right)\cdot y'-3x^{2}y' & = & 6xy\\ | ||
| + | \left(\cos y-3x^{2}\right)y' & = & 6xy\\ | ||
| + | y' & = & \dfrac{6xy}{\cos y-3x^{2}} | ||
| + | \end{array} | ||
| + | \] | ||
Revision as of 20:45, 16 November 2015
Background
So far, you may only have differentiated functions written in the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f\left(x\right)} . But some functions are better described by an equation involving Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} . For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{2}+y^{2}=16} describes the graph of a circle with center Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(0,0\right)} and radius 4, and is really the graph of two functions: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\pm\sqrt{16-x^{2}}} .
Sometimes, functions described by equations in and are too hard to solve for , for example Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{3}+y^{3}=6xy} . This equation really describes 3 different functions of x, whose graph is the curve:
We want to find derivatives of these functions without having to solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} explicitly. We can do this by implicit differentiation, in which we take the derivative of both sides of our equation with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , and do some algebra steps to solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'} (or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dfrac{dy}{dx}} if you prefer), keeping in mind that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is a function of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} throughout the equation.
Example 1
1. Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'} if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin y-3x^{2}y=8} .
[Momentarily think of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=f\left(x\right)} and view the equation as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin\left(f\left(x\right)\right)-3x^{2}\cdot f\left(x\right)=8} to realize that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin\left(f\left(x\right)\right)} term requires the chain rule and the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x^{2}\cdot f\left(x\right)} term needs the product rule when we differentiate with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , while the derivative of 8 is just 0.]
So we get \[ \begin{array}{rcl} \sin y-3x^{2}y & = & 8\\ \left(\cos y\right)\cdot y'-\left(3x^{2}y'+6xy\right) & = & 0\quad\textrm{(derivative of both sides with respect to \ensuremath{x})}\\ \left(\cos y\right)\cdot y'-3x^{2}y' & = & 6xy\\ \left(\cos y-3x^{2}\right)y' & = & 6xy\\ y' & = & \dfrac{6xy}{\cos y-3x^{2}} \end{array} \]