Differentiation

Question 1

d/dx (sin x), d/dx (cos x), d/dx (tan x)

Answer 1

Derivative of sin x = cos x, cos x = -sin x, tan x = sec^2 X ((1/cosx)^2))

Question 2

derivative of u^2-cos u

Answer 2

2u + sin u

Question 3

derivative of exponential funtion =

Answer 3

the exponential function. Der of f(x)=e^x is e^x. so exponential function = its own derivative.

Question 4

derivative of ln (eg f(x)=ln x)

Answer 4

=1/x

Question 5

Derivative product rule

Answer 5

if k(x) = f(x) g(x) then k’(x) = f(x)g’(x) + g(x)f’(x)

Question 6

Derivative quotient rule

Answer 6

if k(x) = f(x)/g(x) then (g(x)f’(x) - f(x)g’(x))/(g(x))^2

Question 7

Chain rule

Answer 7

composite functions, decompose functions first. dy/dx=dy/du du/dx. y is function of u and u is a function of x. In form of operation by something.

Question 8

Chain rule decomposing functions eg y=cos^3 X (if cos x x^3 then just a function product)

Answer 8

1st term something. Replace something with u. Where u is (rest of composite function) (eg with y=sin^2, it is y=sin (something). Set u to equal ‘something’, y=sin (u) where u = x^2 so these are the 2 constituent functions). y=u^3 where u = cos x.
cos^3x same as (cosx)^3 so y=cos x somethimg^3

Question 9

Decomposing functions ctd

Answer 9

y=ln(x^4), ln something so y=ln when u is x^4.

y=sqrt(ln x), sqrt something so y=sqrt u when u=ln x

Question 10

using chain rule

Answer 10

dy/dx=dy/du du/dx. Decompose function then use chain rule. eg y=sin(x^2), y=sin u where u =x^2, by chain rule dy/dx = (cos u) * 2x. (the two derivatives). Express answer in terms of x by substituting x for u from decomposition so here =x^2 so here (cos (x^2)) * 2x

Question 11

Decompose functions without using extra variable u

Answer 11

eg f(x) = cos (4x). Get derivative of first term (here sin). so sin (4x). Then multiply all by der of second term (u term). Here, sin(4x) * 4

Question 12

Differentiating linear expressions

Answer 12

if k(x)=f(ax) and a is constant then k'(x) = af'(ax).  if k(x)=f(ax+b) then k'(x)=af'(ax+b). Eg k(x)=cos(7x+4), so 
7-sin(7x+4), so -7 sin (ax+4)

Question 13

Combining differentiation rules

Answer 13

When more than one rule needed, first rule applied is the one that applies to overall structure. Eg x^2 sin(x^3) as it is a product, do that rule ignoring the fact that there is a composite in 2nd part. eg g(x) = cos (x ln x). -sin(x ln x) d/dx (x ln x) (by chain rule, d/dx shows it is the derivative of (x ln x)). Then get derivative of x ln x. -sin (x ln x) (x * 1/x + (ln x) *1 (product rule)

Question 14

Inverse function rule

Answer 14

dy/dx = 1/(dx/dy). Express x in terms of y, eg y=sin^-1 is x = sin y.So der is cos y therefore 1/cos y. Then express in terms of x using an identity