Question

СРОЧНО!!!!!!Найти f '(1) : 1)f(x)=(x+1)³ (x+2) 2)f(x)=2x-3/5-4x 4)f(x)=2x²/1-7x

Answer 1

Решение на фотографии

Answer 2

Sure, let's find the derivatives of each of the given functions and then evaluate them at x = 1.

1. f(x) = (x+1)³ * (x+2)

To find f'(x), we'll use the product rule: d/dx [u(x)v(x)] = u'(x)v(x) + u(x)v'(x).

Let u(x) = (x+1)³ and v(x) = (x+2). Then, u'(x) = 3(x+1)² and v'(x) = 1.

Now, f'(x) = u'(x)v(x) + u(x)v'(x) f'(x) = 3(x+1)² * (x+2) + (x+1)³ * 1 f'(x) = 3(x+1)² * (x+2) + (x+1)³

Now, let's find f'(1):

f'(1) = 3(1+1)² * (1+2) + (1+1)³ f'(1) = 3(2)² * 3 + 2³ f'(1) = 3(4) * 3 + 8 f'(1) = 12 * 3 + 8 f'(1) = 36 + 8 f'(1) = 44

So, f'(1) = 44.

2. f(x) = (2x - 3)/(5 - 4x)

To find f'(x), we'll use the quotient rule: d/dx [u(x)/v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]².

Let u(x) = 2x - 3 and v(x) = 5 - 4x. Then, u'(x) = 2 and v'(x) = -4.

Now, f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]² f'(x) = [2(5 - 4x) - (2x - 3)(-4)] / (5 - 4x)² f'(x) = (10 - 8x + 8x - 12) / (5 - 4x)² f'(x) = (10 - 12) / (5 - 4x)² f'(x) = -2 / (5 - 4x)²

Now, let's find f'(1):

f'(1) = -2 / (5 - 4(1))² f'(1) = -2 / (5 - 4)² f'(1) = -2 / 1² f'(1) = -2

So, f'(1) = -2.

3. f(x) = 2x² / (1 - 7x)

To find f'(x), we'll use the quotient rule again.

Let u(x) = 2x² and v(x) = 1 - 7x. Then, u'(x) = 4x and v'(x) = -7.

Now, f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)]² f'(x) = [4x(1 - 7x) - 2x²(-7)] / (1 - 7x)² f'(x) = (4x - 28x² + 14x²) / (1 - 7x)² f'(x) = (4x - 14x²) / (1 - 7x)² f'(x) = 2x(2 - 7x) / (1 - 7x)²

Now, let's find f'(1):

f'(1) = 2(1)(2 - 7(1)) / (1 - 7(1))² f'(1) = 2(2 - 7) / (1 - 7)² f'(1) = 2(-5) / (-6)² f'(1) = -10 / 36 f'(1) = -5 / 18

So, f'(1) = -5/18.

To summarize:

1. f'(1) = 44
2. f'(1) = -2
3. f'(1) = -5/18