find the derivative of [(x^2)+1][cosx]

ncert cbse 11th limits and derivatives miscellaneous exercise

23.find the derivative of [(x^2)+1][cosx]

y=[(x^2)+1][cosx]

using product rule of differentiation

y = uv

dy/dx = uv' +vu'

u =[(x^2)+1]

v=cosx

u ' =2x+0 = 2x

v ' = (-sinx)

dy/dx = uv' +vu'

dy/dx = [(x^2)+1](-sinx)) +[cosx][2x]

dy/dx = -(x^2)sinx -sinx +2x cosx

24. find the derivative  of (a(x^2)+sinx )[p+qcosx]

y=[a(x^2)+sinx ] [p+qcosx]

using product rule of differentiation

y = uv

dy/dx = uv' +vu'

u =[a(x^2)+sinx ]

v =[p+qcosx]

u ' = a(2x)+cosx = 2ax+cosx

v ' =0 - qsinx

dy/dx = uv' +vu'

dy/dx = [a(x^2)+sinx ][ - qsinx] + [p+qcosx][ 2ax+cosx]

13. limits and derivatives
miscellaneous exercise

1.

(i)  find the derivative of (-x) using first principles
solution

(ii)   find the derivative of [ (-x)^(-1) ] using first principles

solution

(iii)find the derivative of sin(x+1) using first principles

solution

(iv) find the derivative of cos[x-(pi/8)] using first principles
solution

23.find the derivative of [(x^2)+1][cosx]

solution

24. find the derivative  of (a(x^2)+sinx )[p+qcosx]

solution

28.

find the derivative of x / (1+tanx)

solution

29.

find the derivative of (x+secx)(x-tanx)

solution

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find the derivative of (x+secx)(x-tanx)

ncert cbse 11th limits and derivatives miscellaneous exercise

29.

find the derivative of (x+secx)(x-tanx)

y = (x+secx)(x-tanx)

using product formula for differentiation

y = uv

dy/dx = uv' +vu'

u =(x+secx)

v=(x-tanx)

taking derivative

u' = 1 + secxtanx

v' = 1 - [(secx)^2]

dy/dx = uv' +vu'

dy/dx =(x+secx) {1 - [(secx)^2]} +(x-tanx){1 + secxtanx ]

28.

find the derivative of x / (1+tanx)

using quotient formula for differentiation

y = u/v

dy/dx = { vu' - uv' } / {v^2}

u=x

v=(1+tanx)

taking derivative

u' = 1

v' = 0 + [(secx)^2] =[(secx)^2]

dy/dx = { (1+tanx)(1) - x[(secx)^2] } / {(1+tanx)^2}

dy/dx = { (1+tanx) - x[(secx)^2] } / {(1+tanx)^2}

dy/dx = { vu' - uv' } / {v^2}

13. limits and derivatives
miscellaneous exercise

1.

(i)  find the derivative of (-x) using first principles
solution

(ii)   find the derivative of [ (-x)^(-1) ] using first principles

solution

(iii)find the derivative of sin(x+1) using first principles

solution

(iv) find the derivative of cos[x-(pi/8)] using first principles
solution

28.

find the derivative of x / (1+tanx)

solution

29.

find the derivative of (x+secx)(x-tanx)

solution

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find the derivative of sin(x+1) using first principles

ncert cbse 11th limits and derivatives miscellaneous exercise

1
(iii)find the derivative of sin(x+1) using first principles

f(x) = sin(x+1)

f(x+h) = sin[(x+h) +1]

f(x+h) - f(x) = sin[(x+h) +1] - sin[x+1]

using trigonometry formula trigonometry identities

sinx - siny =2cos[(x+y)/2] sin [(x-y)/2]

f(x+h) - f(x) = sin[(x+h) +1] - sin[x+1]

=2cos{ [(x+h) +1 +x+1 ] /2 } sin{ [(x+h) +1 - (x+1) ] /2}

=2cos{(2x+h+2)/2}sin{h/2}

[f(x+h) - f(x)] / h =[2cos{(2x+h+2)/2}sin{h/2} / h

=cos{(2x+h+2)/2} * [ sin(h/2) / (h/2) ] --------------(1)

using lim [ (sinx)/x ] =1 as x -->0

taking lim as h --> 0 in equation(1)

f ' (x) = cos{(2x+0+2)/2} *[1]

f ' (x) = cos{x+1} cancelling off the 2

(iv) find the derivative of cos[x-(pi/8)] using first principles

f(x) =cos[x-(pi/8)]

f(x+h) =cos[x+h-(pi/8)]

f(x+h) - f(x) = cos[x+h-(pi/8)]  - cos[x-(pi/8)]

 using trigonometry formula trigonometry identities

cosx - cosy = -2sin[(x+y)/2] sin[(x-y)/2]

f(x+h) - f(x) = cos[x+h-(pi/8)]  - cos[x-(pi/8)]

=-2sin{ {[x+h-(pi/8)]+[x-(pi/8)]}/2 }sin{ {[x+h-(pi/8)] - [x-(pi/8)]}/2 }

= (-2)sin{ {2[x-(pi/8)]+h}/2 } sin(h/2)

[f(x+h) - f(x)] / h =[ -2sin{ {2[x-(pi/8)]+h}/2 } sin(h/2) ] / h

= -sin{ {2[x-(pi/8)]+h}/2 }* [ sin(h/2) / (h/2) ] --------------(1)

using lim [ (sinx)/x ] =1 as x -->0

taking lim as h --> 0 in equation(1)

f ' (x)  = - sin[x-(pi/8)]

13. limits and derivatives
miscellaneous exercise

1.

(i)  find the derivative of (-x) using first principles
solution

(ii)   find the derivative of [ (-x)^(-1) ] using first principles

solution

(iii)find the derivative of sin(x+1) using first principles

solution

(iv) find the derivative of cos[x-(pi/8)] using first principles
solution




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There is no guarantee about the data/information on this site. You use the data/information at your own risk. You use the advertisements displayed on this page at your own risk.We are not responsible for the content of external internet sites. Some of the links may not work. Your internet usage may be tracked by the advertising networks  using tracking cookies.

if z = xy / (x-y), show that x² (∂ ²z / ∂x²) + 2xy (∂ ²z / ∂x∂y) + y² (∂ ²z / ∂y²) = 0

.Let u be a function of x, y , z where x , y, z are independent variables and u depends on x, y ,z..When doing partial differentiation w.r.t x, we treat x alone as the independent variable and treat  y and z as constants.

The partial derivative of u with respect to x is usually denoted by  ∂u / ∂x

If ∂u / ∂x is again differentiated partially with respect to x we get the partial derivative denoted as ∂ ²u / ∂x²

If  ∂u / ∂x is again differentiated partially with respect to y we get the partial derivative denoted as ∂ ²u /∂y ∂x

If  ∂u / ∂y is again differentiated partially with respect to x we get the partial derivative denoted as ∂ ²u /∂x ∂y

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collection of problems in integration, differentiation, areas, volumes, trigonometry, matrices, limits

Calculus
problems on integration ------------------------ integration

problems on application of integration like
area, volume, arc length etc ------------------------integration application

problems on differential equations ------differential equation

problems on differentiation------------------------ differentiation

problems on limits ------------------limits

problems on matrices --------------matrices

problems on trigonometry ------trigonometry


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index of math problems


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example of differentiation using quotient rule

find (dy/dx) if y = (x+5)/(x-3)

using quotient rule since y is of u/v form









formulae on differentiation




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