# Exam 1: Derivative as a function / Differentiation

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2.2, 2.3

f '(x)= 1/2

f ' (x)= 3x^2 -3

### 21) Find the derivative of the function using the definition of derivative. g(x)= (sqrt 1+2x)

g '(x)= 1 / [sqrt 1+2x]

### 23) Find the derivative of the function using the definition of derivative. G(t)= 4t / (t+1)

G ' (t) = 4 / (t+1)^2

0

5

3x^2 -4

### Differentiate f(x)= x- 3sinx

f ' (x)= 1 - 3cosx

t^3

(-2/5)x ^ -7/5

4pir^2

5/32*x^4

0

### differentiate y= (x^2+4x+3)/ (sqrt x)

[3/2(sqrt x) + (2/ sqrt x) - 3] / [2x(sqrt x)]

### 21) differentiate v= t^2 - [ (1)/ 4^(sqrt t^3)]

2t + [ (3)/ (4t (4^ sqrt t^3)]

### 23) differentiate z= (A / y^10) + Bcos y

-10A / y^11 - B sin y

### 25) Find equation of tangent line and normal line to the curve at the given point. y= 6cosx, (pi/3, 3)

y= -3(sqrt 3)x + 3 + [pi*(sqrt 3)]. y= [x / (3 sqrt(3)) + 3 - (pi/(9 sqrt(3))

y = 1/2x + 1/2

### Find the first and second derivatives of the function. f(x) = x^4 -3x^3 + 16x

f'(x)= 4x^3 -9x^2 + 16, f ''(x)= 12x^2 - 18x

### Find the first and second derivatives of the function. g(t)= 2 cos t - 3 sin t

g'(t) = -2sin t - 3 cost, g " (t)= -2 cos t + 3 sin t

Example: