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AP Calculus AB: 6.1.2 Finding the Derivative Implicitly

Mathematics14 CardsCreated 8 months ago

This flashcard set explores how to use implicit differentiation to find derivatives of equations not explicitly solved for one variable. It highlights the use of Leibniz notation and the chain rule to differentiate complex expressions involving both variables, and includes practical examples where the derivative depends on both 𝑥 and 𝑦.

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Finding the Derivative Implicitly

  • Leibniz notation is another way of writing derivatives. This notation will be helpful when finding the derivatives of relations that are not functions.

  • Implicit differentiation uses the chain rule in a creative way to find the derivative of functions in implicit form.

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Key Terms

Term
Definition

Finding the Derivative Implicitly

  • Leibniz notation is another way of writing derivatives. This notation will be helpful when finding the derivatives of relations that are no...

note

  • Leibniz notation is another way of writing derivatives.

  • Notice that Leibniz notation can work like an operation,
    instruct...

Given the equation cos^2x+cos^2y=cos^25π, find dy/dx.

dy/dx=−sin x cos x/sin y cos y

Given the equation

tan^2x^2−tan^2y^2=sec5√π,find dx/dy.

dx/dy=y tan y^2 sec^2 y^2/x tan x^2 sec^2 x^2

Given the equation x+y=0,find dy/dx.

dy/dx=−1

Given the equation s^2−t^2=16,find ds/dt.

ds/dt=t/s

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AP Calculus AB: 6.1.2 Finding the Derivative Implicitly
TermDefinition

Finding the Derivative Implicitly

  • Leibniz notation is another way of writing derivatives. This notation will be helpful when finding the derivatives of relations that are not functions.

  • Implicit differentiation uses the chain rule in a creative way to find the derivative of functions in implicit form.

note

  • Leibniz notation is another way of writing derivatives.

  • Notice that Leibniz notation can work like an operation,
    instructing you to find the derivative of something.

  • An explicit equation is an equation in which one of the
    variables is equal to an expression made up of the other
    variable. The equation is explicitly defined in terms of the second variable.

  • Explicit equations usually describe functions.

  • An implicit equation is not organized nicely like an explicit equation. To find the derivative of an implicit equation, you can either solve the equation for one of the variables (putting it in explicit form) or you can use implicit differentiation.

  • Start by taking the derivative of each side of the implicit
    equation with respect to one of the variables. Work each
    term piece by piece.

  • When you encounter a term with a variable different from the one you are differentiating with respect to, treat that variable like a blop and use the chain rule.

  • After you have differentiated each piece, isolate the
    dy/dx-term.

  • Notice that the derivative of y with respect to x is an
    expression containing both x and y. When using the
    derivative, it is important to substitute both the x-value and the y-value into the expression.

Given the equation cos^2x+cos^2y=cos^25π, find dy/dx.

dy/dx=−sin x cos x/sin y cos y

Given the equation

tan^2x^2−tan^2y^2=sec5√π,find dx/dy.

dx/dy=y tan y^2 sec^2 y^2/x tan x^2 sec^2 x^2

Given the equation x+y=0,find dy/dx.

dy/dx=−1

Given the equation s^2−t^2=16,find ds/dt.

ds/dt=t/s

Given the equation 2x+4y=8,find dy/dx.

dy/dx=−1/2

Given the equation x^2+y^2=9,find dy/dx.

dy/dx=−x/y

Given the equation 3x^2+4y^3=7,find dy/dx.

dy/dx=−x/2y^2

Suppose a curve is defined by the equation (x + y)^2 = 4. What is the equation of the line tangent to the curve at (3, −1)?

y = −x + 2

Given the equation x−1−lny=8, find dy/dx.

dy/dx=−y/x^2

Given the equation x+3y=1,find dy/dx.

dy/dx=−1/3

Suppose a curve is defined by the equation 3xy+2(xy)^2+xy^3=1. Find dy/dx.

dy/dx=−3y+4xy^2+y^3/3x+4x^2y+3xy^2

Given the equation sinx^2+siny^2=5,find dy/dx.

dy/dx=−xcosx^2/ycosy^2