y = u^4 and u = 9x2 − x + 7

Question

 y = u⁴ and u = 9x² − x + 7

du/dx = ?

dy/dx = ?

Answer 1

Good practice for the chain rule.

y = u⁴ so by the power rule y'(u) = 4u³

u = 9x² - x + 7 so u'(x) = 18x -1

Since y = [u(x)]⁴ , then y'(x) = y'(u(x)) u'(x) = 4(9x² - x + 7)³ (18x -1) = (72x - 4) (9x² - x + 7)³

Note you cannot multiply outside coefficients into a power.

Answer 2

a) du/dx = 18x-1

b) By applying the chain rule you have dy/dx=dy/du * du/dx.

We already evaluated the second term and we need to evaluate the first one, dy/du.

Since y=u⁴, then dy/du=4u³ which needs to be written in terms of x as 4*(9x²-x+7)³

Therefore, dy/dx=4*(9x²-x+7)³*(18x-1)

Best,

Davide