Question

Найти: f'(2),если f(x)=3x³-6x²-5 f'(1), если f(x)=(x²+3x)√x f'(-1), если f(x)=(2x+3)/(3x+5)

Answer 1

Решение задания смотри на фотографии

Answer 2

To find the derivatives of the given functions and then evaluate them at the specified points, we can use the rules of differentiation. Let's find the derivatives of each function and then substitute the given values.

1. Find f'(x) for f(x) = 3x³ - 6x² - 5: The derivative of a polynomial function f(x) = ax^n is given by f'(x) = n * a * x^(n-1).

For f(x) = 3x³ - 6x² - 5, the derivative f'(x) is: f'(x) = d/dx (3x³ - 6x² - 5) = 3 * 3x^(3-1) - 6 * 2x^(2-1) - 0 = 9x² - 12x

Now, let's evaluate f'(2): f'(2) = 9 * 2² - 12 * 2 = 9 * 4 - 24 = 36 - 24 = 12

So, f'(2) = 12.

2. Find f'(x) for f(x) = (x² + 3x)√x: To differentiate f(x), we'll need to apply the product rule and the chain rule.

f(x) = (x² + 3x)√x = x^(2 + 1/2) + 3x^(1 + 1/2) = x^(5/2) + 3x^(3/2)

Now, the derivative f'(x) is: f'(x) = d/dx (x^(5/2) + 3x^(3/2)) f'(x) = (5/2)x^(5/2 - 1) + (3/2)(d/dx(x^(3/2))) f'(x) = (5/2)x^(3/2) + (3/2)(3/2)x^(3/2 - 1) f'(x) = (5/2)x^(3/2) + (9/4)x^(1/2)

Now, let's evaluate f'(1): f'(1) = (5/2) * 1^(3/2) + (9/4) * 1^(1/2) f'(1) = (5/2) + (9/4) = 10/4 + 9/4 = 19/4

So, f'(1) = 19/4 or 4.75.

3. Find f'(x) for f(x) = (2x + 3)/(3x + 5): To differentiate f(x), we'll use the quotient rule.

f(x) = (2x + 3)/(3x + 5)

Now, the derivative f'(x) is: f'(x) = (d/dx(2x + 3)(3x + 5) - (2x + 3)d/dx(3x + 5)) / (3x + 5)^2 f'(x) = (2(3x + 5) - (2x + 3) * 3) / (3x + 5)^2 f'(x) = (6x + 10 - 6x - 9) / (3x + 5)^2 f'(x) = (1) / (3x + 5)^2

Now, let's evaluate f'(-1): f'(-1) = 1 / (3(-1) + 5)^2 f'(-1) = 1 / (3(-1 + 5))^2 f'(-1) = 1 / (3(4))^2 f'(-1) = 1 / 12^2 f'(-1) = 1 / 144

So, f'(-1) = 1/144.