 Mathematics
20 May, 13:43

# yequals (9 x Superscript 4 Baseline minus 4 x squared plus 6) Superscript 4 To find StartFraction dy Over dx EndFraction , write y as a function of u so that yequals f (u) and uequals g (x). What is uequals g (x) in this case?

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1. 20 May, 14:16
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Step-by-step explanation:

Given the function

y = (9x⁴ - 4x² + 6) ⁴

We need to find the derivative of y with respect to x i. e. dy/dx.

So let u = 9x⁴-4x² + 6

Then y = u²,

Then, y is a function of u, y=f (u)

Also, u is a function of x, u = g (x)

In this case,

u = g (x) = 9x⁴-4x² + 6

So let differentiate this function y (x).

This is a function of a function

Then, we need to find u' (x)

u (x) = 9x⁴-4x² + 6

Then, u' (x) = 36x³ - 8x

Also we need to find y' (u)

Then, y = u²

y' (u) = 2u

Using function of a function formula

dy / dx = dy/du * du/dx

y' (x) = y' (u) * u' (x)

y' (x) = 2u * 36x³ - 8x

y' (x) = 2u (36x³ - 8x)

Since, u = 9x⁴-4x² + 6

Therefore,

y' (t) = 2 (9x⁴-4x² + 6) (36x³ - 8x)

So,

dy/dx = 2 (9x⁴-4x² + 6) (36x³ - 8x)

dy/dx = (18x⁴-8x² + 12) (36x³ - 8x)