Find out the answer to "Use the chain rule to find dz /..." - Plainmath

Rivka Thorpe

Answered question

2021-10-24

Use the chain rule to find dz / ds and dz / dt.

z = x y^{3} - x^{2} y, x = t^{2} + 1, y = t^{2} - 1

Answer & Explanation

Usamah Prosser

Skilled2021-10-25Added 86 answers

We can write

\frac{d z}{d t} = \frac{d z}{d x} \cdot \frac{d x}{d t} + \frac{d z}{d y} \cdot \frac{d y}{d t}

Given that:

z = x y^{3} - x^{2} y, x = t^{2} + 1, y = t^{2} - 1

Find the required derivatives

\frac{d z}{d x} = y^{3} - 2 x y, \frac{d z}{d y} = 3 x y^{2} - x^{2}

\frac{d x}{d t} = 2 t, \frac{d y}{d t} = 2 t

Substitute the expressions

\frac{d z}{d t} = \frac{d z}{d x} \cdot \frac{d x}{d t} + \frac{d z}{d y} \cdot \frac{d y}{d t}

\frac{d z}{d t} = (y^{3} - 2 x y) (2 t) + (3 x y^{2} - x^{2}) (2 t)

\frac{d z}{d t} = 2 t (y^{3} - 2 x y + 3 x y^{2} - x^{2})

Result:

\frac{d z}{d t} = 2 t (y^{3} - 2 x y + 3 x y^{2} - x^{2})

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