Advertisements
Advertisements
Question
Find `"dy"/"dx"` if cos (xy) = x + y
Solution
cos (xy) = x + y
Differentiating both sides w.r.t. x, we get
`-sin(xy)."d"/"dx"(xy) = 1 + "dy"/"dx"`
∴ `-sin(xy)[x"dy"/"dx" + y"d"/"dx"(x)] = 1 + "dy"/"dx"`
∴ `-sin(xy)[x"dy"/"dx" + y xx 1] = 1 + "dy"/"dx"`
∴ `-xsin(xy)"dy"/"dx" - ysin(xy) = 1 + "dy"/"dx"`
∴ `-"dy"/"dx" -sin(xy)"dy"/"dx" = 1 + ysin(xy)`
∴ `-[1 + x sin(xy)]"dy"/"dx" = 1 + ysin(xy)`
∴ `"dy"/"dx" = (-[1 + ysin (xy)])/(1 + x sin(xy)`.
APPEARS IN
RELATED QUESTIONS
If y = eax. cos bx, then prove that
`(d^2y)/(dx^2) - 2ady/dx + (a^2 + b^2)y` = 0
if `y = tan^2(log x^3)`, find `(dy)/(dx)`
Solve the following differential equation:
x2 dy + (xy + y2) dx = 0, when x = 1 and y = 1
If y = log (cos ex) then find `"dy"/"dx".`
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 ex+y = cos(x – y)
Find `"dy"/"dx"` if `e^(e^(x - y)) = x/y`
Find the second order derivatives of the following : `2x^5 - 4x^3 - (2)/x^2 - 9`
Find the second order derivatives of the following : e2x . tan x
Find the second order derivatives of the following : xx
Find `"dy"/"dx"` if, y = `root(3)("a"^2 + "x"^2)`
Find `"dy"/"dx"` if, y = log(10x4 + 5x3 - 3x2 + 2)
Find `"dy"/"dx"` if, y = log(ax2 + bx + c)
Find `"dy"/"dx"` if, y = `5^(("x" + log"x"))`
Fill in the Blank
If 3x2y + 3xy2 = 0, then `"dy"/"dx"` = ________
If `"x"^"m"*"y"^"n" = ("x + y")^("m + n")`, then `"dy"/"dx" = "______"/"x"`
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`.
Find the rate of change of demand (x) of a commodity with respect to its price (y) if y = `(5x + 7)/(2x - 13)`
Find `"dy"/"dx"`, if y = `2^("x"^"x")`.
Differentiate `"e"^("4x" + 5)` with respect to 104x.
If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______
If x = cos−1(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______
If y = cos−1 [sin (4x)], find `("d"y)/("d"x)`
If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x such that the composite function y = f[g(x)] is a differentiable function of x, then `("d"y)/("d"x) = ("d"y)/("d"u)*("d"u)/("d"x)`. Hence find `("d"y)/("d"x)` if y = sin2x
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 ______
State whether the following statement is True or False:
If y = ex, then `("d"y)/("d"x)` = ex
State whether the following statement is True or False:
If x2 + y2 = a2, then `("d"y)/("d"x)` = = 2x + 2y = 2a
State whether the following statement is True or False:
If y = ex, then `("d"^2y)/("d"x^2)` = ex
Find `("d"y)/("d"x)`, if y = (6x3 – 3x2 – 9x)10
`"d"/("d"x) [sin(1 - x^2)]^2` = ______.
Differentiate `sqrt(tansqrt(x))` w.r.t. x
If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`
If y = `sin^-1 {xsqrt(1 - x) - sqrt(x) sqrt(1 - x^2)}` and 0 < x < 1, then find `("d"y)/(dx)`
If f(x) = |cos x|, find f'`((3pi)/4)`
If f(x) = |cos x – sinx|, find `"f'"(pi/6)`
If y = log (cos ex), then `"dy"/"dx"` is:
Differentiate the function from over no 15 to 20 sin (x2 + 5)
If ax2 + 2hxy + by2 = 0, then prove that `(d^2y)/(dx^2)` = 0.
Let f(x) = log x + x3 and let g(x) be the inverse of f(x), then |64g"(1)| is equal to ______.
If f(x) = `{{:(x^3 + 1",", x < 0),(x^2 + 1",", x ≥ 0):}`, g(x) = `{{:((x - 1)^(1//3)",", x < 1),((x - 1)^(1//2)",", x ≥ 1):}`, then (gof) (x) is equal to ______.
If y = em sin–1 x and (1 – x2) = Ay2, then A 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 y = 2x2 + a2 + 22 then `dy/dx` = ______.
Find `"dy"/"dx" if, e ^(5"x"^2- 2"X"+4)`
Find `dy/dx` if, `y=e^(5x^2-2x+4)`
If `y = root5(3x^2 + 8x + 5)^4`, find `dy/dx`
Find `dy/dx` if, y = `e^(5x^2 - 2x + 4)`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
The differential equation of (x - a)2 + y2 = a2 is ______
Find `dy/dx` if, y = `e^(5 x^2 - 2x + 4)`
Find `dy/dx` if ,
`x= e^(3t) , y = e^(4t+5)`
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`.
If y = `tan^-1((6x - 7)/(6 + 7x))`, then `dy/dx` = ______.
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
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^(5x^2-2x+4)`
Solve the following:
If y = `root(5)((3"x"^2 + 8"x" + 5)^4)`, find `"dy"/"dx"`
Find `dy/dx` if, `y = e^(5x^2 - 2x+4)`
If y = `root{5}{(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`
