Question

Use the Chain Rule to find .dt/dz .
z = cos (x+ 7y) ; x = 4t³ ; y = t / 5

Answer 1

The expression for
`\((dz)/(dt)\)` using the Chain Rule is
`\((dz)/(dt) = -\sin(x + 7y) \cdot (d(x + 7y))/(dt)\).` Substituting the given values and their derivatives, we get
`\((dz)/(dt) = -\sin(4t³ + (7t)/(5)) \cdot (12t² + (7)/(5))\).`

Step-by-step explanation:

To find
`\((dz)/(dt)\)` using the Chain Rule, we begin by expressing z as a composite function of t. In this case, (z = cos(x + 7y), where (x = 4t³) and
`(y = (t)/(5)\` ). The Chain Rule states that
`\((dz)/(dt) = (dz)/(dx) \cdot (dx)/(dt) + (dz)/(dy) \cdot (dy)/(dt)\)` . In this context
`, \((dz)/(dx)\) is \(-\sin(x + 7y)\),`
`\((dx)/(dt)\)` is
`\(12t²\)` ,
`\((dz)/(dy)\)` is
`\(-\sin(x + 7y)\)` , and
`\((dy)/(dt)\)` is
`\((7)/(5)\).`

Substituting these values into the Chain Rule formula, we ge
`t \((dz)/(dt) = -\sin(x + 7y) \cdot (d(x + 7y))/(dt)\)` . Expanding further and substituting the derivatives of\(x) and (y), we arrive at the final expression
`\((dz)/(dt) = -\sin(4t³ + (7t)/(5)) \cdot (12t² + (7)/(5))\).`

In conclusion, the application of the Chain Rule allows us to find the rate of change
`\((dz)/(dt)\)` with respect to (t) in a composite function involving z, x, and y. The detailed calculation involves systematically applying the Chain Rule to each component, resulting in the final expression for
`\((dz)/(dt)\).`