Advertisements
Advertisements
प्रश्न
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
उत्तर
Given: x + sin(x + y) = y – cos(x – y)
To Find: Derivative of x + sin(x + y) = y – cos(x – y)
Step-by-step explanation:
Apply the sum/Difference Rule: (f ± g)' = f' ± g'
= `d/(dx) (x) + d/(dx) (sin(x + y)) - d/(dx) (y) + d/(dx) (cos(x - y))`
- `d/(dx) (x)` = 1
- `d/(dx) (sin(x + y)) = cos(x + y) + cos(x + y)((dy)/(dx))`
- `- d/(dx) (y) = - (dy)/(dx)`
- `d/(dx) (cos(x - y))`
Adding up all, we get;
⇒ 0 = `1 + cos(x + y)(1 + d/(dx) (y)) - d/(dx) (y) - sin(x - y)(1 - d/(dx) (y))`
Taking `(dy)/(dx)` on the left-hand side of the equation, we get:
`(dy)/(dx) = (1 + cos(x + y) - sin(x - y))/(- cos(x + y) + 1- sin(x - y))`
Hence, the derivative of the given equation is: `(1 + cos(x + y) - sin(x - y))/(1 - cos(x + y) - sin(x - y))`
APPEARS IN
संबंधित प्रश्न
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `dy/dx` in the following:
2x + 3y = sin y
Find `dy/dx` in the following:
xy + y2 = tan x + y
Find `dy/dx` in the following:
`y = sin^(-1)((2x)/(1+x^2))`
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
Show that the derivative of the function f given by
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Examine the differentialibilty of the function f defined by
\[f\left( x \right) = \begin{cases}2x + 3 & \text { if }- 3 \leq x \leq - 2 \\ \begin{array}xx + 1 \\ x + 2\end{array} & \begin{array} i\text { if } - 2 \leq x < 0 \\\text { if } 0 \leq x \leq 1\end{array}\end{cases}\]
Write the derivative of f (x) = |x|3 at x = 0.
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Let \[f\left( x \right)\begin{cases}a x^2 + 1, & x > 1 \\ x + 1/2, & x \leq 1\end{cases}\] . Then, f (x) is derivable at x = 1, if
Find `dy/dx if x^3 + y^2 + xy = 7`
Discuss extreme values of the function f(x) = x.logx
If `sin^-1((x^5 - y^5)/(x^5 + y^5)) = pi/(6), "show that" "dy"/"dx" = x^4/(3y^4)`
Find `"dy"/"dx"`, if : x = `sqrt(a^2 + m^2), y = log(a^2 + m^2)`
Find `"dy"/"dx"`, if : `x = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
Find `"dy"/"dx"`, if : `x = cos^-1(4t^3 - 3t), y = tan^-1(sqrt(1 - t^2)/t)`.
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Find `"dy"/"dx"` if : x = t2 + t + 1, y = `sin((pit)/2) + cos((pit)/2) "at" t = 1`
Find `"dy"/"dx"` if : x = t + 2sin (πt), y = 3t – cos (πt) at t = `(1)/(2)`
Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.
If x = at2 and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.
If x = cos t, y = emt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.
If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
Find the nth derivative of the following : eax+b
Find the nth derivative of the following : apx+q
Find the nth derivative of the following : sin (ax + b)
Find the nth derivative of the following : cos (3 – 2x)
Find the nth derivative of the following : `(1)/(3x - 5)`
Find the nth derivative of the following : y = eax . cos (bx + c)
If y `tan^-1(sqrt((a - x)/(a + x)))`, where – a < x < a, then `"dy"/"dx"` = .........
Choose the correct option from the given alternatives :
If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Choose the correct option from the given alternatives :
If y = `a cos (logx) and "A"(d^2y)/(dx^2) + "B""dy"/"dx" + "C"` = 0, then the values of A, B, C are
Differentiate the following w.r.t. x : `sin[2tan^-1(sqrt((1 - x)/(1 + x)))]`
Differentiate the following w.r.t. x : `sin^2[cot^-1(sqrt((1 + x)/(1 - x)))]`
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
Differentiate the following w.r.t. x : `cos^-1((sqrt(1 + x) - sqrt(1 - x))/2)`
Differentiate the following w.r.t. x : `tan^-1[sqrt((sqrt(1 + x^2) + x)/(sqrt(1 + x^2) - x))]`
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
If `x = e^(x/y)`, then show that `"dy"/"dx" = (x - y)/(xlogx)`
Differentiate `tan^-1((sqrt(1 + x^2) - 1)/x)` w.r.t. `cos^-1(sqrt((1 + sqrt(1 + x^2))/(2sqrt(1 + x^2))))`
If log y = log (sin x) – x2, show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.
Find `"dy"/"dx"` if, x3 + x2y + xy2 + y3 = 81
Find `"dy"/"dx"` if, xy = log (xy)
Solve the following:
If `"x"^5 * "y"^7 = ("x + y")^12` then show that, `"dy"/"dx" = "y"/"x"`
If log (x + y) = log (xy) + a then show that, `"dy"/"dx" = (- "y"^2)/"x"^2`.
Solve the following:
If `"e"^"x" + "e"^"y" = "e"^((x + y))` then show that, `"dy"/"dx" = - "e"^"y - x"`.
Choose the correct alternative.
If x = `("e"^"t" + "e"^-"t")/2, "y" = ("e"^"t" - "e"^-"t")/2` then `"dy"/"dx"` = ?
If `"x"^7*"y"^9 = ("x + y")^16`, then show that `"dy"/"dx" = "y"/"x"`
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
If x = a t4 y = 2a t2 then `("d"y)/("d"x)` = ______
If y = `sqrt(tansqrt(x)`, find `("d"y)/("d"x)`.
`(dy)/(dx)` of `xy + y^2 = tan x + y` is
Differentiate w.r.t x (over no. 24 and 25) `e^x/sin x`
If 2x + 2y = 2x+y, then `(dy)/(dx)` is equal to ______.
If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... + ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.
Find `dy/dx` at x = 0.
If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
If log (x+y) = log (xy) + a then show that, `dy/dx= (-y^2)/(x^2)`
If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m2y
Find `dy/dx` if, `x = e^(3t), y = e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx = (−y^2)/x^2`
Find `dy/dx"if", x= e^(3t), y=e^sqrtt`
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
Find `dy/(dx) "if" , x = e^(3t), y = e^sqrtt`.
Find `dy/dx` if, `x = e^(3t), y = e^(sqrtt)`
