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1.

Differentiate function with respect to x,f(x)=`(x^(4)+x^(3)+x^(2)+1)/(x)`

Answer» `d/(dx)(x^(4)+x^(3)+x^(2)+1)/x=d/dx(x^(3)+x^(2)+x+1/x)`
`=d/(dx)x^(3) + d/(dx) x^(2)+d/(dx) x+d/(dx)(1/x)`
`=3x^(2)+2x+1-1/x^(2)`
`=(3x^(4)+2x^(3)+x^(2)-1)/(x^(2))`
Discussion
2.

Evaluate `lim_(xtoa) (sinx-sina)/(sqrt(x)-sqrt(a))`

Answer» Given, `underset(xtoa)"lim"(sinx-sina)/(sqrt(x)-sqrt(a)) =underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2)/(sqrt(x)-sqrt(a))`
`[therefore sinC-sinD=2cos(C+D)/2.sin(C-D)/2]`
`=underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2(sqrt(x)+sqrt(a)))/((sqrt(x)-sqrt(a))(sqrt(x)+sqrt(a)))`
`=underset(xto0)"lim"((2cos(x+a))/2sin(x-a)/2 (sqrt(x)+sqrt(a)))/(x-a)`
`=2underset(xto0)"lim"cos(x+a)/(2)(sqrt(x)+sqrt(a)).1/2underset(xto0)"lim"(sin((x-2)/2))/(2((x-a)/2))`
`=2underset(xto0)"lim"cos((x+a)/2)(sqrt(x)+sqrt(a)). 1/2underset(xto0)"lim"(sin((x-a)/2))/((x-a)/2)`
`=2.cosa/2.sqrt(a).1/2` `[therefore underset(xto0)"lim"(sinx)/x=1]`
`=sqrt(a)cosa/2`
Discussion
3.

Evaluate, `lim_(x to 1) (x^(4)-1)/(x-1)=lim_(x to k) (x^(3)-k^(3))/(x^(2)-k^(2))` , then find the value of k.

Answer» Given, `=underset(xto0)"lim"(x^(3)-k^(3))/(x^(2)-k^(2))`
`rArr 4(1)^(4-1)=underset(xtok)"lim"((x^(3)-k^(3))/(x-k))/((x^(2)-k^(2))/(x-k))` `[therefore underset(xtoa)"lim"(x^(n)-a^(n))/(x-a)=na^(n-1)]`
`rArr 4=(underset(xtok)"lim"(x^(3)-k^(3))/(x-k))/(underset(xtok)"lim"(x^(2)-k^(2))/(x-k))` `therefore underset(xtoa)"lim"(f(x))/(g(x))`
`rArr 4=(3k^(2))/(2k) rArr 4=3/2k`
`therefore k=(4 xx2)/(3)=8/3`
Discussion
4.

Evaluate: `lim_(x->1) (x^15-1)/(x^10-1)`

Answer» numerator,`x^15 - 1`
`(x^5)^3 - 1`
`= (x^5-1)[(x^5)^2 + x^5 + 1]`
`= (x^5 - 1)[x^10 + x^5 +1]`
denominator,. `x^10 - 1 = (x^5)^2 - 1`
`= (x^5 - 1)(x^5+1)`
`lim_(x->1) ((x^5 -1){x^10 + x^5 +1})/((x^5 - 1)(x^5+1))`
`lim_(x->1) (x^10 + x^5 +1)/x^5`
`= (1^10 + 1^5+ 1)/(1^5 +1)= 3/2`
`lim_(x->1) (x^15 - 1)/(x^10 - 1)`
`= lim_(x->1) (d/dx(x^15-1))/(d/dx(x^10-1))`
`lim_(x->1) (15x^14)/(10 x^9)`
`= 15/10 = 3/2`
Answer
Discussion
5.

Find the derivative of the following functions (it is to be understand that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): `(4x+5sinx)/(3x+7cos x)`

Answer» Let `y=(4x+5 sinx)/(3x+7cos x)`
`rArr(dy)/(dx)=(d)/(dx)[((4x+5sinx)/(3x+7cos x)]`
`(3x+7cos x)(d)/(dx)(4x+5 sinx)`
`=(-(4x+5sinx)(d)/(dx)(3x+7cos x))/((3x+7 cos x)^(2))`
`(12x+15x cos x+28 cos x+35cos^(2)x)`
`=(-(12x-28xsinx+15sinx-35sin^(2)x)+28x sin x-15 sin x)/((3x+7 cos x)^(2))`
`15x cos x +28 cos x +35`
`=(15x cos x+28 cos x+35+28xsinx-15sinx)/((3x+7cos x)^(2))`
Discussion