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| Calculus problems on differentiation --------------------------------------------------------
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partial differentiationdiff. arctan √ [(1-x)/(1+x)]
w.r.t.x ---------------diff. by
trig. substitution
derivative of (ax+b)/(cx+d) --------------quotient rule find dy/dt if y = t exp(-t) [Acost +Bsint] -----------uvw rule find dy/dx if y= x² tanֿ¹(5x) ---------product (uv) rule implicit
differentiation
if sqrt(x) +sqrt(y) = 8, find
dy/dx ----------->( implicit
differentiation )
if √(xy) = x - 2y, find dy/dx ---------------->( implicit differentiation ) if xy + y^2 =1 , find dy/dx ---------------- implicit differentiation if y = x^(lnx) , find dy/dx ----------------logarthmic differentiation if siny = x sin(a+y) show that dy/dx = sin²(a+y) / sina ----------answer explanation differentiation from first principles
------------------------------------ * * ------------------------------------ differentiate sqrt(x) from first
principles -----> (differentiation by first principles)
-------------------------------------------------------------find the derivative of f(x) =1/x at x=3 using the lim {f(x)-f(a)} / {x-a} as x---->a -----------first principle example on chain rule
differentiate ln(x + sqrt(x^2-1))
w.r.t.x -----> ( derivative of ln(x +
sqrt(x^2-1)))
-------------------------------------------------------------example on chain rule
find the derivative
of arccos[ (1- x²) / (1 + x²) ] w.r.t
arctan(x)----------find
the derivative of one funtion w.r.t. another
-------------------------------------------------------------rate of
change
lamppost-shadow problem ------>( lamppost-shadow problem) -------------------------------------------------------------
maxima
minima
-------------------------------------------------------------show that the semivertical angle of a right circular cone of maximum volume and given slant height is tan ֿ¹(√2 ) explanation of maxima / minima problem show that the height of a closed cylinder of given volume and minimum surface area is equal to its diameter explanation of minimising surface area A piece of string 28m long is to be cut into two pieces, one piece is to be made into a circle and the other the boundary of a square. How should the string be cut if the sum of the areas of the two figures is to be a minimum answer and explanation There is a figure (norman window) in which a rectangle is surmounted by a semicircle with diameter along one side of the rectangle .If the perimeter is given find the radius of the semicircle if the area is to be maximum (maximum amount of light is to be admitted into the room) explanation tangent
find the equation of the tangent
at (1,1) on x²+y²+xy-3=0 ----tangent on a
curve
equation of the tangent at (1,1) on x² +xy+2y² = 4 -----------equation of tangent using calculus -------------------------------------------------------------
problem
on mean value theorem
show that (b-a) / (1
+b²) < arctan(b) - arctan(a)
< (b-a) / (1 +a²) if a<b explanation
-------------------------------------------------------------example on maclaurin's series
maclaurin's series for sec(x)
-----> (maclaurin's series
for sec(x))
-------------------------------------------------------------power series for arctan(x) -----> (power series for tan ֿ¹(x)) if u =ln{sqrt (x ² +y ² )} , find ∂u/∂x ,∂u/∂y,∂²u/∂x² ,∂²u/∂x∂y ---------partial differentiation index of problems with some classification --------------index |
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