is a powerful tool for finding derivatives of complex functions that aren't easily expressed in terms of y = f(x). It allows us to tackle equations where x and y are intertwined, like circles or other curves.
This technique is crucial for solving related rates problems and finding tangent slopes of implicit curves. By differentiating both sides of an equation and using the chain rule, we can uncover hidden relationships between variables and their rates of change.
Implicit Differentiation
Implicit differentiation for complex functions
Top images from around the web for Implicit differentiation for complex functions
Implicit Differentiation – Math Mistakes View original
Is this image relevant?
Implicit Differentiation Homework Set - Wisewire View original
Is this image relevant?
The Product and Quotient Rules - Wisewire View original
Is this image relevant?
Implicit Differentiation – Math Mistakes View original
Is this image relevant?
Implicit Differentiation Homework Set - Wisewire View original
Is this image relevant?
1 of 3
Top images from around the web for Implicit differentiation for complex functions
Implicit Differentiation – Math Mistakes View original
Is this image relevant?
Implicit Differentiation Homework Set - Wisewire View original
Is this image relevant?
The Product and Quotient Rules - Wisewire View original
Is this image relevant?
Implicit Differentiation – Math Mistakes View original
Is this image relevant?
Implicit Differentiation Homework Set - Wisewire View original
Is this image relevant?
1 of 3
Technique used to find derivative when function not explicitly defined as y=f(x)
Function defined implicitly by equation relating x and y (x2+y2=25)
Steps to find derivative using implicit differentiation:
Differentiate both sides of equation with respect to x, treating y as function of x
Apply chain rule when differentiating terms involving y, multiplying by dxdy
Solve resulting equation for dxdy to find derivative
Examples of implicit differentiation:
Equation x2+y2=25, differentiating both sides yields 2x+2ydxdy=0
Product rule and quotient rule are often used in implicit differentiation when dealing with more complex equations
Tangent slopes of implicit curves
Slope of tangent line to curve at given point equals value of derivative at that point
Steps to find slope of tangent line for implicitly defined curve:
Use implicit differentiation to find derivative dxdy
Substitute coordinates of point of interest into derivative to calculate slope
Example:
Curve defined by x2+y2=25, find slope of tangent line at point (3, 4)
Derivative found using implicit differentiation: dxdy=−yx
Substituting point (3, 4) into derivative gives dxdy=−43, slope of tangent line is −43
Curve defined by xy=10, find slope of tangent line at point (2, 5)
Derivative: y+xdxdy=0, solving for dxdy gives dxdy=−xy
Substituting point (2, 5) yields slope of −25
Related rates via implicit differentiation
Related rates problems involve finding rate of change of one quantity with respect to another
Given relationship between quantities and rate of change of one quantity
Steps to solve related rates problems:
Identify given information and quantity to be found
Write equation relating quantities using given information
Differentiate both sides of equation with respect to time, using implicit differentiation if necessary
Substitute known values and solve for desired rate of change
Example problem:
Circular pool being filled with water at rate of 5 ft³/min. Find rate at which water level is rising when water is 3 ft deep.
Volume of cylinder (pool) is V=πr2h, where r is radius and h is height (water depth)
Differentiating both sides with respect to time gives dtdV=2πrdtdrh+πr2dtdh
Given dtdV=5 ft³/min and h=3 ft, solve for dtdh to find rate at which water level is rising
Ladder 13 ft long leaning against wall, bottom sliding away at rate of 2 ft/sec. Find rate at which top is sliding down wall when bottom is 5 ft from wall.
Pythagorean theorem relates quantities: x2+y2=132, where x is distance from wall and y is height on wall
Differentiate: 2xdtdx+2ydtdy=0
Substitute x=5, dtdx=2, and y=132−52=12 to solve for dtdy
Additional Concepts in Differentiation
Differentiation is the process of finding the derivative of a function
The derivative represents the rate of change of a function at any given point
Tangent lines to a curve can be found using the derivative, as the slope of the tangent line equals the derivative at that point
Key Terms to Review (2)
Folium of Descartes: The folium of Descartes is a planar algebraic curve defined by the equation $x^3 + y^3 - 3axy = 0$. It has a characteristic loop and intersects itself at the origin.
Implicit differentiation: Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. It involves differentiating both sides of an equation with respect to the independent variable and then solving for the desired derivative.