USING CHAIN RULE WITH SQUARE ROOT POLYNOMIALS

Find the derivative:

Problem 1 :     

y =∜(7m³ - 4m² + 2)

Solution :

y = ∜(7m³ - 4m² + 2)

y = (7m³ - 4m²  + 2)^1/4, then g(m) = 7m³ - 4m² + 2

Let f(m) = m^1/4

f’(m) = (1/4) m^-3/4

f’(m) = 1/4m^3/4

g(m) = 7m³ - 4m² + 2

g’(m) = 21m² - 8m

dy/dm = f’(g(m)) g’(m)

= [ 1/4(7m³ - 4m² + 2)^3/4 ] · 21m² - 8m

dy/dm = (21m² - 8m) / 4 ∜ (7m³ - 4m² + 2)³

Problem 2 :

c = ∛ (4k² + 3)²

Solution :  

c = ∛ (4k² + 3)²

c = (4k² + 3)^2/3

Let f(k) = k^2/3, then g(k) = 4k² + 3

f’(k) = (2/3) k^-1/3

f’(k) = 2/3k^1/3

g(k) = 4k² + 3

g’(k) = 8k

dc/dk = f’(g(k)) g’(k)

= [2/3(4k² + 3)^1/3] · 8k

dc/dk = 16k / 3∛ (4k² + 3)

Problem 3 :

r = ∜ (4w + 3)⁵

Solution :

r = ∜ (4w + 3)⁵

r = (4w + 3)^5/4

Let f(w) = w^5/4, g(w) = 4w + 3

f’(w) = (5/4) w^1/4

g(w) = 4w + 3

g’(w) = 4

dr/dw = f’(g(w)) g’(w)

= 5/4 (4w + 3)^1/4 · 4

dr/dw = 5 ∜ (4w + 3)

Problem 4:

d = 3 / √x + 2

Solution:  

d = 3 / √x + 2

d = 3 (x + 2)^-1/2

dd/dx = 3(-1/2) (x + 2)^-3/2 · d/dx(x + 2)

= -3/2 d/dx(x + 2) / (x + 2)^3/2

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

= -3/2(x + 2)^3/2

dd/dx = -3/2√(x + 2)³

Problem 5 :

f = 2 / √4e + 5

Solution :  

f = 2 / √4e + 5

f = 2 (4e + 5)^-1/2

df/de = 2 (-1/2)(4e + 5)^-1/2-1 · d/de (4e + 5)

= -4 d/de (e + 5) / (4e + 5)^3/2

= -4/(4e + 5)^3/2

df/de = -4/√(4e + 5)³

Problem 6 :

g(x) = 1/√1 - 2x

Solution :  

g(x) = 1/√1 - 2x

g(x) = 1 (1 - 2x)^-1/2

g’(x) = (-1/2)(1 - 2x)^-3/2 · d/dx (1 - 2x)

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

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

g’(x) = 1/√(1 - 2x)³

Problem 7 :

k(n) = 4/√n² + 6

Solution :  

k(n) = 4/√n² + 6

k(n) = 4 (n² + 6)^-1/2

k’(n) = 4(-1/2) (n² + 6)^-3/2 d/dn (n² + 6)

= -2(2n + 0) / (n² + 6)^3/2

k’(n) = -4n / √(n² + 6)³

Problem 8  :

p(r) = 12 / ∜ (7 - r)⁵

Solution :  

p(r) = 12 / ∜ (7 - r)⁵

p(r) = 12 (7 - r)^-5/4

p’(r) = 12(-5/4) (7 - r)^-9/4 d/dr (7 - r)

= -15(0 - 1) / (7 - r)^9/4

p’(r) = 15 / ∜ (7 - r)⁹

Problem 9 :

q(z) = 5 / ⁵√ (z⁵ - 32)

Solution :  

q(z) = 5 / ⁵√ (z⁵ - 32)

q(z) = 5 (z⁵ - 32)^-1/5

q’(z) = 5 (-1/5) (z⁵ - 32)^-6/5 d/dz (z⁵ - 32)

=   - (5z⁴ - 0) / (z⁵ - 32)^6/5

q’(z) = -5z⁴ / ⁵√ (z⁵ - 32)⁶