Advertisements
Advertisements
Question
Find `"dy"/"dx"`, if y = xx.
Solution
y = xx.
Taking logarithm of both sides, we get
log y = log (xx)
∴ log y = x log x
Differentiating both sides w.r.t.x, we get
`1/"y" * "dy"/"dx" = "x" * "d"/"dx" (log "x") + log "x" * "d"/"dx" ("x")`
`= "x" * 1/"x" + log "x" (1)`
∴ `1/"y" * "dy"/"dx" = 1 + log x`
∴ `"dy"/"dx" = "y"(1 + log "x")`
∴ `"dy"/"dx" = "x"^"x" (1 + log "x")`
APPEARS IN
RELATED QUESTIONS
If y = eax. cos bx, then prove that
`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0
Find `"dy"/"dx"` if `sqrt(x) + sqrt(y) = sqrt(a)`
Find `"dy"/"dx"` if cos (xy) = x + y
Find the second order derivatives of the following : e2x . tan x
Find the second order derivatives of the following : e4x. cos 5x
Find `"dy"/"dx"` if, y = `root(3)("a"^2 + "x"^2)`
Find `"dy"/"dx"` if, y = log(log x)
Find `"dy"/"dx"` if, y = log(ax2 + bx + c)
If y = 2x2 + 22 + a2, then `"dy"/"dx" = ?`
Differentiate `"e"^("4x" + 5)` with respect to 104x.
Find `("d"y)/("d"x)`, if y = `root(5)((3x^2 + 8x + 5)^4`
If u = x2 + y2 and x = s + 3t, y = 2s - t, then `(d^2u)/(ds^2)` = ______
Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`
If f(x) = |cos x|, find f'`((3pi)/4)`
If y = log (cos ex), then `"dy"/"dx"` is:
If ax2 + 2hxy + by2 = 0, then prove that `(d^2y)/(dx^2)` = 0.
Find `dy/dx` if, y = `e^(5 x^2 - 2x + 4)`
Find `dy/dx` if, y = `e^(5x^2-2x+4)`
Solve the following:
If y = `root5((3x^2 +8x+5)^4`,find `dy/dx`
If `y = (x + sqrt(a^2 + x^2))^m`, prove that `(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - m^2y = 0`
