Advertisements
Advertisements
प्रश्न
If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`
Advertisements
उत्तर
x sin(a + y) + sin a cos(a + y) = 0 .......(i)
`x.cos("a" + y)*"d"/("d"x)("a" + y) + sin("a" + y)*"d"/("d"x)(x) + sin"a"[-sin("a" + y)]*"d"/("d"x)("a" + y)` = 0
∴ `xcos("a" + y )("d"y)/("d"x) + sin("a" + y)(1) - sin("a" + y)("d"y)/("d"x)` = 0
∴ `[x cos("a" + y) - sin "a" sin("a" + y)]("d"y)/("d"x)` = − sin(a + y) .......(ii)
From (i), we get
x = `(-sin"a"cos("a" + y))/(sin("a" + y))`
Substituting the value of x in (ii), we get
`[(-sin"a"cos("a" + y))/(sin("a" + y))*cos("a" + y) - sin"a"sin("a" + y)]("d"y)/("d"x)` = − sin(a + y)
∴ `-sin"a"[(cos^2("a" + y))/(sin("a" + y)) + sin("a" + y)]("d"y)/("d"x)` = − sin(a + y)
∴ `(-sin"a"[cos^2("a" + y) + sin^2("a" + y)])/(sin("a" + y))("d"y)/("d"x)` − sin(a + y)
∴ `-(sin"a"(1))/(sin("a" + y))*("d"y)/("d"x)` = − sin(a + y)
∴ `("d"y)/("d"x) = sin("a" + y) [(sin("a" + y))/(sin"a")]`
∴ `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`
संबंधित प्रश्न
Find `"dy"/"dx"`, if x = at2, y = 2at
Find `(dy)/(dx)`, if x = 2at2, y = at4.
Find `"dy"/"dx"`, if x = e3t, y = `"e"^((4"t" + 5))`
Find `"dy"/"dx"`, if x = `("u" + 1/"u")^2, "y" = (2)^(("u" + 1/"u"))`
Find `"dy"/"dx"`, if x = `sqrt(1 + "u"^2), "y" = log (1 + "u"^2)`
Find `"dy"/"dx"`, if Differentiate 5x with respect to log x
Solve the following.
If x = `"a"(1 - 1/"t"), "y" = "a"(1 + 1/"t")`, then show that `"dy"/"dx" = - 1`
If x = t . log t, y = tt, then show that `dy/dx - y = 0`.
If x = 2at2 , y = 4at, then `dy/dx = ?`
If x = `y + 1/y`, then `dy/dx` = ____.
Find `"dy"/"dx"` if x = 5t2, y = 10t.
Choose the correct alternative:
If x = 2am, y = 2am2, where m be the parameter, then `("d"y)/("d"x)` = ?
State whether the following statement is True or False:
If x = 2at, y = 2a, where t is parameter, then `("d"y)/("d"x) = 1/"t"`
State whether the following statement is True or False:
If x = 5m, y = m, where m is parameter, then `("d"y)/("d"x) = 1/5`
Find `("d"y)/("d"x)`, if x = em, y = `"e"^(sqrt("m"))`
Solution: Given, x = em and y = `"e"^(sqrt("m"))`
Now, y = `"e"^(sqrt("m"))`
Diff.w.r.to m,
`("d"y)/"dm" = "e"^(sqrt("m"))("d"square)/"dm"`
∴ `("d"y)/"dm" = "e"^(sqrt("m"))*1/(2sqrt("m"))` .....(i)
Now, x = em
Diff.w.r.to m,
`("d"x)/"dm" = square` .....(ii)
Now, `("d"y)/("d"x) = (("d"y)/("d"m))/square`
∴ `("d"y)/("d"x) = (("e"sqrt("m"))/square)/("e"^"m")`
∴ `("d"y)/("d"x) = ("e"^(sqrt("m")))/(2sqrt("m")*"e"^("m")`
If x = `sqrt(1 + u^2)`, y = `log(1 + u^2)`, then find `(dy)/(dx).`
If x = f(t) and y = g(t) are differentiable functions of t, then prove that:
`dy/dx = ((dy//dt))/((dx//dt))`, if `dx/dt ≠ 0`
Hence, find `dy/dx` if x = a cot θ, y = b cosec θ.
Find `dy/dx` if, x = e3t, y = `e^((4t+5))`
Find `dy/dx` if, x = `e^(3t)`, y = `e^(4t+5)`
If x = f(t) and y = g(t) are differentiable functions of t, so that y is function of x and `(dx)/dt ≠ 0` then prove that `dy/(dx) = (dy/dt)/((dx)/dt)`. Hence find `dy/(dx)`, if x = at2, y = 2at.
Find `dy/dx if,x = e^(3^T), y = e^((4t + 5)`
Find `dy/dx` if x= `e^(3t)`, y =`e^((4t+5))`
Find `dy/dx` if, `x = e^(3t), y = e^((4t + 5))`
Find `dy/dx if, x= e^(3t)"," y = e^((4t+5))`
Find `dy/dx` if, x = `e^(3t)`, y = `e^((4t + 5))`.
