Calculus Derivative Homework help 10 points!!?

Question

f(x)=4x^7-3x^5+4x^4-x^3+root 3 

g(t)=4t^(2)-3t^(-2)+6 

z=2root(t)-4sin(t) 

h(x)=(3x-5)(4x^2+2) 

y=2x^2-5/(2x^2-3x+3) 

y=3x^3e^x 

Answer 1

The function f(x) = 4x⁷ - 3x⁵ + 4x⁴ - x³ + √3

Apply dervative on each side with respect of x.

Apply the formulae d/dx( xⁿ ) = nx^{n -1}

d/dx(constant) = 0

f ' (x) = 28x⁶  - 15x⁴  + 16x³  - 3x² + 0

f ' (x) = 28x⁶  - 15x⁴  + 16x³  - 3x²

Answer 2

The function g(t) = 4t²  - 3t^-2  + 6

Apply dervative on each side with respect of t.

g'(t) = 8t + 6t^-3

 

The function z = 2√t - 4sin(t)

Apply the formula d/dx (√x) = 1/(2√x)

d/dx(sinx) = cosx

z' = 2[1/(2√t)] - 4cos(t)]

z' = 1/√t - 4cos(t)

 

Answer 3

The function h(x) = (3x - 5) (4x² + 2)

Apply derivative on each side with respect of x.

Apply product rule in derivatives d/dx(uv) = uv' + vu'.

u = 3x - 5, v = 4x² + 2

u' = 3, v' = 8x

h'(x) = (3x - 5)(8x) + (4x² + 2)(3)

= 24x² - 40x + 12x² + 6

h'(x) = 36x²  - 40x + 6.

 

The function y = (2x² - 5)/ (2x² - 3x + 3)

Apply derivative on each side with respect of x.

Apply quotient rule in derivatives d/dx(u/v) = [vu' - uv']/v² .

u = 2x² - 5, v = 2x² - 3x + 3

u' = 4x , v' = 4x - 3

y' = [ (2x² - 3x + 3)(4x) - (2x² - 5)(4x - 3)]/(2x² - 3x + 3)²

y' = [ 8x³ - 12x² + 12x - 8x³ + 20x + 6x² - 15]/(2x² - 3x + 3)²

y' = [ - 6x² + 32x - 15 ]/(2x² - 3x + 3)².

Answer 4

The function y = 3x³ e^x

Apply dervative on each side with respect of x.

Apply product rule in derivatives d/dx(uv) = uv' + vu'.

u = 3x³ , v = e^x

u' = 9x² v' = e^x

y' = (3x³)(e^x) + (e^x)(9x²)

y' = 3x²e^x [ x + 3 ].