Advertisements
Advertisements
Question
If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`
Advertisements
Solution
Given that: x sin (a + y) + sin a cos (a + y) = 0
⇒ x sin (a + y) = – sin a cos (a + y)
⇒ x = `(-sin"a" * cos("a" + y))/(sin ("a" + y))`
⇒ x = – sin a.cot (a + y)
Differentiating both sides w.r.t. y
⇒ `"dx"/"dy" = - sin"a"*"d"/"dy" cot("a" + y)`
⇒ `"dx"/"dy" = -sin"a"[-"cosec"^2("a" + y)`
⇒ `"dx"/"dy" = sin"a"/(sin^2("a" + y))`
∴ `"dy"/"dx" = 1/("dx"/"dy")`
= `1/(sin"a"/(sin^2("a" + y))`
Hence, `"dy"/"dx" = (sin^2("a" + y))/sin"a"`.
Hence proved.
APPEARS IN
RELATED QUESTIONS
If x = a sin t and `y = a (cost+logtan(t/2))` ,find `((d^2y)/(dx^2))`
If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`
Find the second order derivative of the function.
e6x cos 3x
Find the second order derivative of the function.
tan–1 x
Find the second order derivative of the function.
log (log x)
Find the second order derivative of the function.
sin (log x)
If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.
If y = cos–1 x, find `(d^2y)/dx^2` in terms of y alone.
If y = 3 cos (log x) + 4 sin (log x), show that x2y2 + xy1 + y = 0.
If y = 500e7x + 600e–7x, show that `(d^2y)/(dx^2)` = 49y.
Find `("d"^2"y")/"dx"^2`, if y = `sqrt"x"`
Find `("d"^2"y")/"dx"^2`, if y = `"x"^-7`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^"log x"`
Find `("d"^2"y")/"dx"^2`, if y = `"e"^((2"x" + 1))`.
Find `("d"^2"y")/"dx"^2`, if y = `"x"^2 * "e"^"x"`
`sin xy + x/y` = x2 – y
Derivative of cot x° with respect to x is ____________.
Let for i = 1, 2, 3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let,
A(x) = `[(p_1(x), p_1^'(x), p_1^('')(x)),(p_2(x), p_2^'(x), p_2^('')(x)),(p_3(x), p_3^'(x), p_3^('')(x))]`
and B(x) = [A(x)]T A(x). Then determinant of B(x) ______
If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.
Find `(d^2y)/dx^2 if, y = e^((2x + 1))`
Find `(d^2y)/dx^2` if, `y = e^((2x + 1))`
Find `(d^2y)/(dx^2)` if, y = `e^((2x+1))`
Find `(d^2y)/dx^2 "if," y= e^((2x+1))`
Find `(d^2y)/(dx^2) "if", y = e^((2x + 1))`
Which expression defines the second order derivative of \[y\] with respect to \[x\]?
If \[y=\mathrm{A}\sin x+\mathrm{B}\cos x\], what is \[\frac{dy}{dx}\]?
If \[y=\sin^{-1}x\], what is \[\frac{dy}{dx}\]?
Which equation is equivalent to \[\frac{dy}{dx}=\frac{1}{\sqrt{1-x^2}}\] for \[y=\sin^{-1}x\]?
What equation is obtained after multiplying through by \[\sqrt{1-x^2}\] in the derivation for \[y=\sin^{-1}x\]?
