Advertisements
Advertisements
प्रश्न
Find `"dy"/"dx"`if, y = `(1 + 1/"x")^"x"`
उत्तर
y = `(1 + 1/"x")^"x"`
Taking logarithm of both sides, we get
log y = `log(1 + 1/"x")^"x"`
∴ log y = x `log(1 + 1/"x")`
Differentiating both sides w.r.t.x, we get
`1/"y" * "dy"/"dx" = "x" * "d"/"dx" log(1 + 1/"x") + log(1 + 1/"x") * "d"/"dx" ("x")`
∴ `1/"y" * "dy"/"dx" = "x" * 1/(1 + 1/"x") * "d"/"dx" (1 + 1/"x") + log (1 + 1/"x") * (1)`
∴ `1/"y" * "dy"/"dx" = "x"/(("x" + 1)/"x") * (0 - 1/"x"^2) + log (1 + 1/"x")`
∴ `1/"y" * "dy"/"dx" = "x"^2/("x + 1") * ((-1)/"x"^2) + log (1 + 1/"x")`
∴ `1/"y" * "dy"/"dx" = (- 1)/("x + 1") + log (1 + 1/"x")`
∴ `"dy"/"dx" = "y"[(-1)/("x + 1") + log (1 + 1/"x")]`
∴ `"dy"/"dx" = (1 + 1/"x")^"x" * [log (1 + 1/"x") - 1/("x + 1")]`
APPEARS IN
संबंधित प्रश्न
Find `"dy"/"dx"`if, y = `"x"^("e"^"x")`
Find `"dy"/"dx"`if, y = `root(3)(("3x" - 1)/(("2x + 3")(5 - "x")^2))`
Fill in the blank.
If x = t log t and y = tt, then `"dy"/"dx"` = ____
If y = x log x, then `(d^2y)/dx^2`= _____.
Fill in the blank.
If y = `"e"^"ax"`, then `"x" * "dy"/"dx" =`____
The derivative of ax is ax log a.
Find `"dy"/"dx"` if y = `sqrt(((3"x" - 4)^3)/(("x + 1")^4("x + 2")))`
If u = ex and v = loge x, then `("du")/("dv")` is ______
Find `("d"y)/("d"x)`, if y = [log(log(logx))]2
Find `("d"y)/("d"x)`, if y = xx + (7x – 1)x
Solve the following differential equations:
x2ydx – (x3 – y3)dy = 0
If y = (log x)2 the `dy/dx` = ______.
Find `dy/dx` if, y = `x^(e^x)`
Find `dy/dx if, y = x^(e^x)`
Find `dy / dx` if, `y = x^(e^x)`
Find `dy/dx "if", y = x^(e^x)`
Find `dy/dx` if, `y = x^(e^x)`
Find `dy/(dx) "if", y = x^(e^(x))`
