Advertisements
Advertisements
рдкреНрд░рд╢реНрди
If `y = (x + sqrt(a^2 + x^2))^m`, prove that `(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - m^2y = 0`
рдЙрддреНрддрд░ рез
`y = [x + sqrt(a^2 + x^2)]^m`
`dy/dx = (m[x + sqrt(a^2 + x^2)]^m)/(x + sqrt(a^2 + x^2))*[1 + (1*2x)/(2sqrt(a^2 + x^2))]`
= `my 1/sqrt(a^2 + x^2)`
`sqrt(a^2 + x^2)dy/dx = my`
`sqrt(a^2 + x^2)(d^2y)/(dx^2) + dy/dx * 1/2 * (2x)/sqrt(a^2 + x^2) = mdy/dx`
`(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - msqrt(a^2 + x^2)dy/dx = 0`
`(a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - m^2y = 0`
рдЙрддреНрддрд░ реи
`y = [x + sqrt(a^2 + x^2)]^m`
`dy/dx = (m[x + sqrt(a^2 + x^2)]^m)/(x + sqrt(a^2 + x^2)]*[1 + (1.2x)/(2sqrt(a^2 + x^2))]`
= `my 1/sqrt(a^2 + x^2)`
`sqrt(a^2 + x^2)dy/dx = my`
Squaring both sides we get,
`(a^2 + x^2)(dy/dx)^2 = m^2y^2`
Differentiating w.r.t ‘ЁЭСе’,
`2x(dy/dx)^2 + (a^2 + x^2)2dy/dx*(d^2y)/(dx^2) = 2m^2ydy/dx`
`\implies (a^2 + x^2)(d^2y)/(dx^2) + xdy/dx - m^2y = 0`
APPEARS IN
рд╕рдВрдмрдВрдзрд┐рдд рдкреНрд░рд╢реНрди
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 the second order derivatives of the following : xx
Find `"dy"/"dx"` if, y = `root(3)("a"^2 + "x"^2)`
If x = cos−1(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______
Suppose y = f(x) is a differentiable function of x on an interval I and y is one – one, onto and `("d"y)/("d"x)` ≠ 0 on I. Also if f–1(y) is differentiable on f(I), then `("d"x)/("d"y) = 1/(("d"y)/("d"x)), ("d"y)/("d"x)` ≠ 0
If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost
If y = `1/sqrt(3x^2 - 2x - 1)`, then `("d"y)/("d"x)` = ?
Choose the correct alternative:
If y = `root(3)((3x^2 + 8x - 6)^5`, then `("d"y)/("d"x)` = ?
If y = x2, then `("d"^2y)/("d"x^2)` is ______
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
If f(x) = `(x - 2)/(x + 2)`, then f(α x) = ______
If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`
If y = log (cos ex), then `"dy"/"dx"` is:
y = `sec (tan sqrt(x))`
y = `2sqrt(cotx^2)`
If ax2 + 2hxy + by2 = 0, then prove that `(d^2y)/(dx^2)` = 0.
If `y = root5(3x^2 + 8x + 5)^4`, find `dy/dx`
Solve the following:
If y = `root5((3x^2 + 8x + 5)^4)`, find `dy/dx`
lf 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 prove that:
`dy/dx = dy/(du) xx (du)/dx`
Hence, find `d/dx[log(x^5 + 4)]`.
Solve the following:
If y = `root5((3x^2 +8x+5)^4`,find `dy/dx`
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 = f(u) is a differentiable function of u and u = g(x) is a differentiate function of x such that the composite function y = f[g(x)] is a differentiable function of x then prove that
`dy/dx = dy/(du) xx (du)/dx`
Hence find `dy/dx` if y = log(x2 + 5)
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)`
Find `dy/dx` if, `y = e^(5x^2 - 2x + 4)`.
