4. Applications of Derivatives

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

y² = x²(4 − x) / 4 + x (right strophoid)

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

y² = x²(4 − x) / 4 + x (right strophoid)

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

y² = x²(4 − x) / 4 + x (right strophoid)

A challenging derivative Find dy/dx, where √3x⁷+y² = sin²y+100xy.

A challenging second derivative Find d²y/dx², where √y+xy=1.

Evaluate and simplify y'.

x = cos (x−y)

Evaluate and simplify y'.

xy⁴+x⁴y=1

Evaluate and simplify y'.

sin x cos(y−1) = 1/2

Let f(x) = x².

a. Show that f(x)−f(y) / x−y = f′(x+y²), for all x≠y.

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

cos y = x; (0, π/2)

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

cos y = x; (0, π/2)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

tan xy = x+y; (0,0)

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

tan xy = x+y; (0,0)

13-26 Implicit differentiation Carry out the following steps.

a. Use implicit differentiation to find dy/dx.

³√x+³√y⁴ = 2;(1,1)