Advertisements
Advertisements
Question
Find `"dy"/"dx"`, if y = xx.
Advertisements
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
Find `dy/dx if x + sqrt(xy) + y = 1`
Find `dy/dx if x^2y^2 - tan^-1(sqrt(x^2 + y^2)) = cot^-1(sqrt(x^2 + y^2))`
Find `"dy"/"dx"` if `e^(e^(x - y)) = x/y`
Find `"dy"/"dx"` if, y = log(log x)
Find `"dy"/"dx"` if, y = `"a"^((1 + log "x"))`
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`
If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______
Choose the correct alternative:
If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?
If y = (5x3 – 4x2 – 8x)9, then `("d"y)/("d"x)` is ______
If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.
If y = log (cos ex), then `"dy"/"dx"` is:
y = sin (ax+ b)
y = `cos sqrt(x)`
If ax2 + 2hxy + by2 = 0, then prove that `(d^2y)/(dx^2)` = 0.
Let x(t) = `2sqrt(2) cost sqrt(sin2t)` and y(t) = `2sqrt(2) sint sqrt(sin2t), t ∈ (0, π/2)`. Then `(1 + (dy/dx)^2)/((d^2y)/(dx^2)` at t = `π/4` is equal to ______.
Let f(x) = x | x | and g(x) = sin x
Statement I gof is differentiable at x = 0 and its derivative is continuous at that point.
Statement II gof is twice differentiable at x = 0.
If `d/dx` [f(x)] = ax+ b and f(0) = 0, then f(x) is equal to ______.
Find `"dy"/"dx" if, e ^(5"x"^2- 2"X"+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"`
Solve the following:
If`y=root(5)((3x^2+8x+5)^4),"find" (dy)/dx`
Find `"dy"/"dx"` if, `"y" = "e"^(5"x"^2 - 2"x" + 4)`
Find the rate of change of demand (x) of acommodity with respect to its price (y) if
`y = 12 + 10x + 25x^2`
If f(x) = `sqrt(7*g(x) - 3)`, g(3) = 4 and g'(3) = 5, find f'(3).
If x = Φ(t) is a differentiable function of t, then prove that:
`int f(x)dx = int f[Φ(t)]*Φ^'(t)dt`
Hence, find `int(logx)^n/x dx`.
If y = `log((x + sqrt(x^2 + a^2))/(sqrt(x^2 + a^2) - x))`, find `dy/dx`.
Find `dy/dx` if, `y=e^(5x^2-2x+4)`
Find `(dy) / (dx)` if, `y = e ^ (5x^2 - 2x + 4)`
If `y = root{5}{(3x^2 + 8x + 5)^4}, "find" dy/dx`.
