Question

Differentiate with respect to t

Assume x = x(t)
Assume y = y(t)
Assume z = z(t)

x^2 - y^3 + z^4 = 1

Answer 1

`\displaystyle 2x(dx)/(dt)-3y^2(dy)/(dt)+4z^3(dz)/(dt)=0`

Explanation:

We want to differentiate the equation:

`x^2-y^3+z^4=1`

With respect to t, where x, y, and z are functions of t.

So:

`\displaystyle (d)/(dt)\left[x^2-y^3+z^4\right]=(d)/(dt)\left[1\right]`

Implicitly differentiate on the left. On the right, the derivative of a constant is simply zero. Hence:

`\displaystyle 2x(dx)/(dt)-3y^2(dy)/(dt)+4z^3(dz)/(dt)=0`