Advertisements
Advertisements
प्रश्न
Find the nth derivative of the following:
y = e8x . cos (6x + 7)
Advertisements
उत्तर
y = e8x . cos (6x + 7)
∴ `"dy"/"dx" = "d"/"dx"[e^(8x).cos (6x + 7)]`
= `e^(8x)."d"/"dx"[cos (6x + 7)] + cos (6x + 7)."d"/"dx"(e^(8x))`
= `e^(8x).[-sin(6x + 7)]."d"/"dx"(6x + 7) + cos(6x + 7).e^(8x)."d"/"dx"(8x)`
= – e8x sin (6x + 7) x (b x 1 + 0) + e8xcos(6x + 7) x a x 1
= e8x [a cos (6x + 7) – b sin (6x + 7)]
= `e^(8x).sqrt(a^2 + b^2)[a/sqrt(a^2 + b^2)cos(6x + 7) - b/sqrt(a^2 + b^2)sin(6x + 7)]`
Let `a/sqrt(a^2 + b^2) = cos x and b/sqrt(a^2 + b^2) = sin x`
Then tan ∞ = `b/a`
∴ ∞ = `tan^-1(b/a)`
∴ `"dy"/"dx" = e^(8x).sqrt(a^2 + b^2)[cosoo.cos(bx + c) - sinoo.sin(bx + c)]`
= `e^(8x).(a^2 + b^2)^(1/2).cos(6x + 7 + x)`
`(d^2y)/(dx^2) = "d"/"dx"[e^(8x).(a^2 + b^2)^(1/2).cos(6x + 7 + oo)]`
= `(a^2 + b^2)^(1/2)."d"/"dx"[e^(8x).cos(6x + 7 + oo)]`
= `(a^2 + b^2)^(1/2)[e^(8x)."d"/"dx"{cos(6x + 7 + oo)} + cos(6x + 7 + oo)."d"/"dx"(e^(8x))]`
= `(a^2 + b^2)^(1/2)[e^(8x).{-sin(6x + 7 + oo)}."d"/"dx"(6x + 7 + oo) + cos(6x + 7 + oo).e^(8x)."d"/"dx"(8x)]`
= `(a^2 + b^2)^(1/2)[-e^(8x)sin(6x + 7 + oo) xx (b xx 1 + 0 + 0) + cos(6x + 7 + oo).e^(8x) xx a xx 1]`
= `e^(8x).(a^2 + b^2)^(1/2)[a cos (6x + 7 + oo) - bsin(6x + 7 + oo)]`
= `e^(8x).(a^2 + b^2)^(1/2).sqrt(a^2 + b^2)[a/sqrt(a^2 + b^2)cos(6x + 7 + oo) = b/sqrt(a^2 + b^2)sin(6x + 7 + oo)]`
= `e^(8x).(a^2 + b^2)^(2/2)[cosoo.cos(6x + 7 + ∞) - sinoo.sin(6x + 7 + oo)`
= `e^(8x).(a^2 + b^2)^(2/2).cos(6x + 7 + oo + oo)`
= `e^(8x).(a^2 + b^2)^(2/2).cos(6x + 7 + 2oo)`
Similarly.
`(d^3y)/(dx^3) = e^(8x).(a^2 + b^2)^(3/2).cos(6x + 7 + 3oo)`
In general, the nth order derivative is given by
`(d^ny)/(dx^n) = e^(8x).(a^2 + b^2)^(n/2).cos(6x + 7 + noo)`,
Where ∞ = `tan^-1(b/a)`
∴ `(d^ny)/(dx^n) = e^(8x).(10)^n.cos[6x + 7 + ntan^-1(3/4)]`
APPEARS IN
संबंधित प्रश्न
If xpyq = (x + y)p+q then Prove that `dy/dx = y/x`
Find `bb(dy/dx)` in the following:
2x + 3y = sin x
Find `bb(dy/dx)` in the following:
sin2 x + cos2 y = 1
if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`
If \[f\left( x \right) = x^3 + 7 x^2 + 8x - 9\]
, find f'(4).
Find the derivative of the function f defined by f (x) = mx + c at x = 0.
Is |sin x| differentiable? What about cos |x|?
If \[\lim_{x \to c} \frac{f\left( x \right) - f\left( c \right)}{x - c}\] exists finitely, write the value of \[\lim_{x \to c} f\left( x \right)\]
Differentiate e4x + 5 w.r..t.e3x
Find `(dy)/(dx) , "If" x^3 + y^2 + xy = 10`
If x = tan-1t and y = t3 , find `(dy)/(dx)`.
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 = cos^-1((2t)/(1 + t^2)), y = sec^-1(sqrt(1 + t^2))`
Find `"dy"/"dx"` if : x = cosec2θ, y = cot3θ at θ= `pi/(6)`
If x = `(t + 1)/(t - 1), y = (t - 1)/(t + 1), "then show that" y^2 + "dy"/"dx"` = 0.
Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`
Differentiate xx w.r.t. xsix.
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 y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.
If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show" (d^2y)/(dx^2)` = 0.
If x2 + 6xy + y2 = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.
If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.
Find the nth derivative of the following:
`(1)/x`
Find the nth derivative of the following : cos (3 – 2x)
Choose the correct option from the given alternatives :
If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?
Differentiate the following w.r.t. x : `tan^-1((sqrt(x)(3 - x))/(1 - 3x))`
If sin y = x sin (a + y), then show that `"dy"/"dx" = (sin^2(a + y))/(sina)`.
Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).
Find `"dy"/"dx"` if, `"x"^"y" = "e"^("x - y")`
Find `"dy"/"dx"` if, xy = log (xy)
If log (x + y) = log (xy) + a then show that, `"dy"/"dx" = (- "y"^2)/"x"^2`.
Choose the correct alternative.
If y = 5x . x5, then `"dy"/"dx" = ?`
Choose the correct alternative.
If ax2 + 2hxy + by2 = 0 then `"dy"/"dx" = ?`
State whether the following is True or False:
The derivative of `"x"^"m"*"y"^"n" = ("x + y")^("m + n")` is `"x"/"y"`
If x2 + y2 = t + `1/"t"` and x4 + y4 = t2 + `1/"t"^2` then `("d"y)/("d"x)` = ______
Find `(dy)/(dx)`, if `y = sin^-1 ((2x)/(1 + x^2))`
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
If y = y(x) is an implicit function of x such that loge(x + y) = 4xy, then `(d^2y)/(dx^2)` at x = 0 is equal to ______.
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`
Solve the following.
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`.
If log(x + y) = log(xy) + a then show that, `dy/dx = (-y^2)/x^2`
