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Question
Find `"dy"/"dx"` if x = `"e"^"3t", "y" = "e"^(sqrt"t")`.
Solution
x = `"e"^"3t"`
Differentiating both sides w.r.t. t, we get
`"dx"/"dt" = "e"^"3t" * "d"/"dt" ("3t")`
`= "e"^"3t" * (3)`
∴ `"dx"/"dt" = 3"e"^"3t"`
y = `"e"^(sqrt"t")`
Differentiating both sides w.r.t. t, we get
`"dy"/"dt" = "e"^(sqrt"t") * "d"/"dt" (sqrt"t")`
`"dy"/"dt" = "e"^(sqrt"t") * 1/(2 sqrt"t")`
∴ `"dy"/"dx" = ("dy"/"dt")/("dx"/"dt") = "e"^(sqrt"t")/((2 sqrt"t")/(3"e"^"3t"))`
∴ `"dy"/"dx" = 1/(6 sqrt"t") "e"^(sqrt"t" - "3t")`
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