Question

Y=3(2x²-x)² знайти похідну функції

Answer 1

$y=3(2x^2-x)^2\\y'=3*2(2x^2-x)*(4x-1)=6(2x^2-x)(4x-1)$

Answer 2

To find the derivative of the given function Y = 3(2x² - x)² with respect to x, we'll use the chain rule and power rule of differentiation.

Let's break down the process step by step:

Step 1: Identify the inner function u and the outer function v.

In this case, the inner function u is (2x² - x), and the outer function v is 3u².

Step 2: Find the derivative of the inner function u with respect to x (du/dx).

The derivative of u = (2x² - x) is given by: du/dx = d(2x²)/dx - d(x)/dx = 2(2x) - 1 = 4x - 1

Step 3: Find the derivative of the outer function v with respect to u (dv/du).

The derivative of v = 3u² with respect to u is: dv/du = d(3u²)/du = 3 * 2u^(2-1) = 6u

Step 4: Apply the chain rule.

The chain rule states that if we have a composition of functions (v ∘ u)(x), then the derivative of this composition with respect to x is given by (dv/du) * (du/dx).

So, the derivative of Y with respect to x is: dY/dx = (dv/du) * (du/dx) = 6u * (4x - 1)

Step 5: Substitute the value of u.

We know that u = (2x² - x), so: dY/dx = 6(2x² - x) * (4x - 1)

Now, simplify the expression: dY/dx = 6(8x³ - 4x² - 2x² + x) dY/dx = 6(8x³ - 6x² + x)

The final derivative of the given function Y = 3(2x² - x)² with respect to x is: dY/dx = 6(8x³ - 6x² + x)