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Question

Consider the following.

3x⁴ + y⁷ = 5x

(a) Find y ′ by implicit differentiation.

y ′ = ?

(b)Solve the equation explicitly for y and differentiate to get y ′ in terms of x.

y ′ = ?

(c)Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part (a).

y ′ = ?

Answer 1

3x⁴ + y⁷ = 5x

12x³ + 7y⁶ (dy/dx) = 5

dy/dx = (5 - 12x³) / (7y⁶)

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y⁷ = 5x - 3x⁴

y = (5x - 3x⁴)^1/7

dy/dx = (1/7)(5x - 3x⁴)^-6/7(5 - 12x³)

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dy/dx = (5 - 12x³) / (7((5x - 3x⁴)^1/7)⁶) = (1/7)(5 - 12x³)(5x - 3x⁴)^-6/7

^

Answer 2

a) 3x^4 + y^7 = 5x => 12x^3dx + 7y^6dy = 5dx => 7y^6dy = 5dx - 12x^3dx => 7y^6dy = (5-12x^3)dx

=> y' = dy/dx = (5-12x^3)/(7y^6).

b) 3x^4 + y^7 = 5x => y^7 = 5x - 3x^4 => y = (5x-3x^4)^(1/7) => y' = (1/7)(5x-3x^4)^(-6/7)(5-12x^3)

=> (1/7)(5-12x^3)/(5x-3x^4)^(6/7).

c) Use y = (5x-3x^4)^(1/7) to substitute for y on a)'s y' = (5-12x^3)/(7y^6) to get (5-12x^3)/(7(5x-3x^4)^(6/7)) which is equivalent to the answer found in b).