Differentiation
Differentiate the following:
y=7x3/5 + 7/5x3 + 4x/3 -4/3cbrt(x5)
dy/dx = 21x2/5 - 21/5x4 +4/3 + 20/9cbrt(x8)
Differentiate 3cos3x + x
-9sin3x+1
Let f(x) = 5x2+10. Find the equation of the tangent at P(1,15).
y =10x + 5
Consider the function f(x)= x3 + 3x2 -9x which passes through the origin. Find any points of inflexion and justify your answer.
x=-1; check by table of signs
Integrate (x+1)(x+2)
x3/3 - 3x2/2 +x+c
Let f(x) = x3 - 2x2 - 1
Find f'(x)
find the gradient of the curve at (2,-1)
f'(x)=3x2 - 4x
f'(2)=4
Use the quotient rule to prove that the derivative of tan(x) is 1/(cos(x))2
tan x = sinx / cos x
dy/dx = sinx.sinx - cos x(-cos x)/ (cos x)2
=1/(cos x)2
Find the equation of the tangent line and the equation of the normal to the curve with equation y = x3 + 1 at the point (1,2).
T: y=3x-1
N: y= -1/3. x + 7/3
Differentiate each of the functions below and hence determine whether it is increasing or decreasing on its entire domain.
f(x)= x3 + x +5
f(x) = 5-3e2x
f(x) = (2x+1)/(3x+5)Increasing; decreasing; decreasing
Let f'(x)=6x2+1. Find f(x) given f(0) = 3
2x3+3
f(x)= 3x2/(5x-1)
Write down the equation of the vertical asymptote
Find f'(x)
x= 1/5
f'(x) = (15x2-6x)/(5x-1)2
Differentiate ln(x2 + 1) + 2cos((pi/2)x)
(2x/(x2+1))-pi.sin((pi/2)x)
Consider the function f (x) = 4x3 + 2x . Find the equation of the normal to the curve of f at the point where x =1 .
y=-1/14. x +85/14
Consider the function f(x)= x3 + 3x2 -9x which passes through the origin. Find any stationary points and determine their nature.
x=-3 max
x=-1 min
Integrate 4x-3 -12x-2 +6x+ 3
-2/x2 +12/x + 6ln(x)+3x+c
Consider the function f (x) = x ln(x) - x .
Find f'(x)
Find the coordinate of of the point such that the gradient is 1
ln(x)
(0,e)
Find the derivative of y=5x2cos(3x2).
10x(cos(3x2)-3x2sin(3x2))
Find the equations of the tangent line and the normal line to curve y = (x -1)4
(a) at point P(0,1).
((b) at x=1
a. y =-4x +1, y= 1/4.x +1
b. At x =1, y = 0 , mt= 0 mn not defined.
Tangent: y = 0 , Normal x =1
Let g(x) = ln(x)/x2 for x>0. Find g'(x). The graph of g has a maximum point at A. Find the x-coordinate of A.
g'(x)= (1-2ln(x))/x3)
x=e1/2; y=0.184
Given that the integral between k and 3 of 1/(x-2) .dx = ln (7) find the value of k .
k=9
The n -th derivative of a function f is denoted by f(n)(x)
(a) Let f (x) = x4 . Find the first four derivatives of f (f(4)(x)
(b) Let g(x) = x4 +ax3 +bx2 +cx + d . Write down the value of g(4) (x) .
(c) Let h(x)= xm . Find the value of g(m)(x) in terms of m .
(d) Let k(x)=1/x; find k(4)(x). Guess the n -th derivative of a function k
1
k(x)
x
(i) Show that (4)
5
24
k (x)
x
. [5]
(ii) Guess a formula for ( ) ( ) n k x , the n -th derivative of k(x)
a. 24; b. 24; c. m!; d. 24/x5 ; e. n!/xn+1
If f(x) = ln(2x-1), x>1/2 find f'(x) and the value of x where the gradient of f (x) is equal to x .
f'(x)= 2/(2x-1)
1.28
Let f (x) = kx4 . The point P(1, k) lies on the curve of f . At P, the normal to the curve is parallel to y = -1/8. x. Find the value of k.
k=2
Consider the function h(x) = (x-2)/(x-1)2, x cannot = 1. The graph has a point P that is a point of inflection. Find h'(x). given h''(x) = (2x-8)/(x-1)4, calculate the coordinates of P
h'(x) = (3-x)/(x-1)3
P(4,2/9)
Consider the curves y = x2 and 8-x2. Find the area A enclosed by the two curves and the x-axis in the first quadrant.
4.42