6.1.2 Finding the Derivative Implicitly

Question 1

Finding the Derivative Implicitly

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

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

Answer 3

dy/dx=−sin x cos x/sin y cos y

Question 4

Given the equation

tan^2x^2−tan^2y^2=sec5√π,find dx/dy.

Answer 4

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

Question 5

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

Answer 5

dy/dx=−1

Question 6

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

Answer 6

ds/dt=t/s

Question 7

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

Answer 7

dy/dx=−1/2

Question 8

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

Answer 8

dy/dx=−x/y

Question 9

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

Answer 9

dy/dx=−x/2y^2

Question 10

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)?

Answer 10

y = −x + 2

Question 11

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

Answer 11

dy/dx=−y/x^2

Question 12

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

Answer 12

dy/dx=−1/3

Question 13

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

Answer 13

dy/dx=−3y+4xy^2+y^3/3x+4x^2y+3xy^2

Question 14

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

Answer 14

dy/dx=−xcosx^2/ycosy^2