Question

find the total derivatives for each of the following functins: 6u^2+15uy+3y^2 , Where y=7u^2

Answer 1

The function f (u ) = 6u ² + 15u y  + 3y ².

where y = 7u ²

Substiute the y  value in the function.

f (u ) = 6u ² + 15u (7u ² ) + 3(7u ² ) ²

f (u ) = 6u ² + 105u ³ + 3(49u⁴)

f (u ) = 6u ² + 105u ³ + 147u⁴

Apply derivative on each side with respect of u .

d /du [f (u )] = d /du [6u ² + 105u ³ + 147u⁴]

f  ' (u ) = d /du (6u ²) + d /du (105u ³ ) + d /du (147u⁴)

Apply the formula d /dx (x ⁿ ) = n x ^{n  - 1}

f  ' (u ) =  12u + 315u ² + 588u ³

Therefore, f  ' (u ) =  588u ³ + 315u ² + 12u.