Implicit Differentiation

Implicit Differentiation

Multiple Choice

Warm Up

Find the derivative f(t) = (t² + 2t)⁵

• f'(t) = 5(2t+2)⁴
• f'(t) = 5(t² + 2t)⁴
• f'(t) = 5(t² + 2t)⁴(2t + 2)

Multiple Select

Which of the following equations, if any, are written in implicit form?

(Check all that apply)

• $$y=4\ x-9$$

• $$x^{\ 2}+y^{\ 2}=49$$

• $$x\ y=16$$

• $$y=\sqrt{36-x^{\ 2}}$$

• $$y=\frac{25\ x}{x^{\ 2}-64}$$

Multiple Choice

Differentiate with respect to x

$$y\ ^7$$  

• $$7y\ ^6\times\frac{\text{d}y}{\text{d}x}$$  

• $$8y\ ^7$$  

• $$7y^6$$  

• $$8y^{7\ }\times\frac{\text{d}y}{\text{d}x}$$  

Multiple Choice

Differentiate w.r.t.x

$$x^2+y^2+2$$  

• $$2x+2y\frac{\text{d}y}{\text{d}x}+2$$  

• $$2x+2y\frac{\text{d}y}{\text{d}x}$$  

• $$2x+2y$$  

Example:

Multiple Choice

Find  $$\frac{dy}{dx}$$ :  $$5x^2\ =3y^2+1$$  

• $$\frac{5x}{3y}$$  

• $$\frac{3y}{5x}$$  

• $$\frac{4x}{3y}$$  

• $$\frac{2y}{3x}$$  

Multiple Choice

Find  $$\frac{dy}{dx}$$ :  $$y^2=10x$$  

• $$\frac{5}{y}$$  

• $$\frac{10}{y}$$  

• $$\frac{y^2}{5}$$  

• $$\frac{y^2}{10}$$  

Multiple Choice

Find dy/dx by Implicit Differentiation 
x³ +y³  = 36

• 6 -x
• 3x² +3y² 
• $$-\frac{x^2}{y^2}$$  

• 0

Multiple Choice

Find   $$\frac{dy}{dx}$$  :  $$2x^2-3y^2=4$$  

• $$\frac{2x}{y}$$  

• $$\frac{x}{3y}$$  

• $$\frac{3y}{2x}$$  

• $$\frac{2x}{3y}$$  

Multiple Choice

Find the derivative of $$x^2+xy+y^3=0$$  

• $$-\frac{x+3y^2}{2x+y}$$  

• $$-\frac{2x+y}{x+3y^2}$$  

• $$-\frac{2x}{x+3y^2}$$  

Multiple Choice

Find dy/dx
xy+y²=2

• -y/(x+2y)
• y/(x+2y)
• -3y/x
• -3x/y

Multiple Choice

Find dy/dx at the given point

x³ +2xy -y²=11 at (2,3)

• -4/7

• 12

• -9

• 9

Multiple Choice

Find the slope of the tangent line at the given point

x³ +2xy -y²=11 at (2,3)

• -4/7

• 12

• -9

• 9

Related Rates

Poll

Warm Up

The United Arab Emirates just launched its first ever mission to Mars

depend to the equation

$$x^2+y^3-2y=3$$  

, find the slope of the tangent line at the point (2,1).

• -4

• 4

• -8

• 8

​Remember

Sphere Volume​

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​  Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.

Multiple Choice

A spherical snowball is rolled in fresh snow, causing it to grow so that its radius increases at a rate of 4 in/sec. How fast is the volume of the snowball increasing when the radius is 9 in?

• 1296π in³/sec

• 1294π in³/sec

• 1303π in³/sec

• 1305π in³/sec