Differentiate the equation 𝑥^7+5𝑥𝑦^3=5 with respect to the variable

Question

Differentiate the equation 𝑥^7+5𝑥𝑦^3=5 with respect to the variable t and express 𝑑𝑦/𝑑𝑡 in terms of 𝑑𝑥/𝑑𝑡. (Use D for dx/dt).

dy/dt =

Answer 1

You use the chain rule to perform this differentiation. Think of "x" as a function having the variable "t" imbedded in it. You are unaware of what the derivative of "x" with respect to "t" is (because you don't know the exact function rule) so you just call the derivative "dx/dt". So, for instance, for the first term, x⁷, the derivative is 7x⁶•(the derivative of "x" with respect to "t") or 7x⁶•(dx/dt).

For the 2nd term, 5xy³, use the product rule and chain rule. The product rule is (u•v)' = u'v + uv' where u = 5x and v = y³ so u' = 5•dx/dt and v' = 3y²•dy/dt

So the derivative of 5xy³ with respect to t is (5•dx/dt)(y³) + (5x)•(3y²•dy/dt)

The derivative of the constant 5 is zero so taking the derivative of x⁷ + 5xy³ = 5 with respect to "t" gives us:

7x⁶•(dx/dt) + (5•dx/dt)(y³) + (5x)•(3y²•dy/dt) = 0

then substituting D = dx/dt gives:

7x⁶•D + (5•D)(y³) + (5x)•(3y²•dy/dt) = 0

7Dx⁶ + 5Dy³ + 15xy²•dy/dt = 0

15xy²•dy/dt = -7Dx⁶ - 5Dy³

dy/dt = (-7Dx⁶ - 5Dy³)/(15xy²)