HOW TO FIND THE DERIVATIVE OF A SQUARE ROOT FUNCTION

Calculate dy/dx for the following functions:

Problem 1 :

y = (x - 3) √(x - 3)

Solution :

y = (x - 3) √(x - 3)

y = (x - 3) (x - 3)^1/2

y = (x - 3)^3/2

dy/dx = 3/2 (x - 3)^(3/2-1) d/dx (x - 3)

= 3/2 (x - 3)^1/2 (1 - 0)

dy/dx = 3√(x - 3)/2

Problem 2 :

y = √(4 - √(x + 2))

Solution :

y = √(4 - √(x + 2))

Problem 3 :

y = (7x + √x² + 3)⁶

Solution : 

 y = (7x + √x² + 3)⁶

Problem 4 :

f(x) = √(2x +5)/(7x - 9)

Solution : 

Problem 5 :

f(x) = √(5x³ - 2x)

Solution : 

f(x) = √(5x³ - 2x)

Problem 6 :

y = 3x² √(4x² - 5x + 1)

Solution : 

y = 3x² √(4x² - 5x + 1)

Problem 7 :

y = (2/3)t³ √(3t³ - 5t)

Solution : 

y = (2/3)t³ √(3t³ - 5t)

= (2/3)t³[(9t² - 5)/2√(3t³ - 5t)] + √(3t³ - 5t) (2/3) (3t²)

= [t³(9t² - 5)/3√(3t³ - 5t)] + 2t²√(3t³ - 5t)

Problem 8 :

y = 2e^x √x

Solution : 

y = 2e^x √x

u = e^x    v = √x

u' = e^x    v' = 1/2√x

dy/dx = (e^x)(1/2√x) + √xe^x

Factoring e^x

dy/dx = (e^x)[(1/2√x) + √x]

Problem 9 :

e^{√(2x^3 - x^2)}

Solution :

Problem 10 :

8√x⁵ - 7x⁴ + 5/√x

Solution :