differentiate below functions

Question

1) y= (t^3+t)/(t^4-3)

2)f(x)=cos(8x+2)

3)f(x)= cos(sin(x^2))

Answer 1

1)

y = (t³ + t) / (t⁴ - 3)

Diferenciate each side with respective t

dy / dt = d / dt((t³ + t) / (t⁴ - 3))

Recall : Diferenciate formula d(u / v) = (vu' - uv') / v^2

substitute u = t³ + t , v = t⁴ - 3 , u' = 3t² + 1 and v' = 4t³ in the above diferecial formula

dy / dt = [(t⁴ - 3)(3t² + 1) - (t³ + t)(4t²)]  /  (t⁴ - 3)²

dy / dt = [3t⁶ + t⁴ - 9t²  - 3 - 4t⁵ + 4t³] / (t⁴ - 3)^2

dy / dt = [3t⁶ - 4t⁵ + t⁴ + 4t³ - 9t²  - 3] / (t⁴ - 3)^2.

Comments

dy/dt = [(t⁴ - 3)(3t² + 1) - (t³ + t)(4t³)]  /  (t⁴ - 3)²

dy/dt = [3t⁶ - 9t² + t⁴ - 3 - (4t⁶ + 4t⁴)]  /(t⁴ - 3)²

dy/dt = [3t⁶ - 9t² + t⁴ - 3 - 4t⁶ - 4t⁴)]  /(t⁴ - 3)²

dy/dt = [- t⁶ - 9t² - 3t⁴ - 3]  /(t⁴ - 3)²

dy/dt = - ( t⁶ + 3t⁴ + 9t² + 3]  /(t⁴ - 3)²

Answer 2

2)

f '(x) = cos(8x + 2)

Diferenciate with respective x

Recall :Diferenciate formula d / dx(cos(ax + b)) = -asin(ax + b)

f '(x) = -8sin(8x + 2).

 

 

Answer 3

3)

f(x) = cos(sin(x²)

sin(x^2) = t then dt = cos(x²) 2xdx

f(x) = cos(t)

Diferenciate each side with respective t

f ' (x)dx = -sin(t) dt

Substitute t = sin(x²) and dt = cos(x²) 2xdx in the above f ' (x)

f ' (x) dx = -sin(sin(x²)) cos(x²) 2x dx

f '(x) = -sin(sin(x²)) cos(x²) 2x.