Key to "Use the Chain Rule to find dz /..." - Plainmath

Dottie Parra

Answered question

2021-10-24

Use the Chain Rule to find dz / ds and dz / dt.

z = (x - y) 5, x = s^{2} t, y = s t^{2}

Answer & Explanation

Adnaan Franks

Skilled2021-10-25Added 92 answers

Finding

\frac{d z}{d t}

and finding

\frac{d z}{d s}

We can write

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

Let us fint the necessary partial derivates

\frac{d z}{d x} = \frac{d [{(x - y)}^{5}]}{d x} = 5 {(x - y)}^{4}

\frac{d z}{d y} = \frac{d [{(x - y)}^{5}]}{d y} = - 5 {(x - y)}^{4}

\frac{d x}{d s} = \frac{d [s^{2} t]}{d s} = 2 s t

\frac{d y}{d s} = \frac{d [s t^{2}]}{d s} = t^{2}

Substitute the expressions

\frac{d z}{d s} = 5 {(x - y)}^{4} \cdot 2 s t - 5 {(x - y)}^{4} \cdot t^{2}

\frac{d z}{d s} = 5 {(x - y)}^{4} (2 s t - t^{2})

Result:

\frac{d z}{d s} = \frac{d z}{d t} = 5 {(x - y)}^{4} (2 s t - t^{2})

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