Question

F(x)=2√x *(x+1) f`(4)= f(x)=x+1/x^2 f`(2)=

Answer 1

$1)\ \ f(x)=2\sqrt{x}\cdot (x+1)\ \ \to \ \ f(x)=2x^{3/2}+2\sqrt{x}\\\\f'(x)=2\cdot \dfrac{3}{2}\, x^{1/2}+2\cdot \dfrac{1}{2\sqrt{x}}=3\sqrt{x}}+\dfrac{1}{\sqrt{x}}\\\\f'(4)=3\cdot 2+\dfrac{1}{2}=6,5$

$2)\ \ a)\ \ f(x)=x+\dfrac{1}{x^2}\ \ ,\qquad \ \ \Big(\dfrac{C}{u}\Big)'=\dfrac{-C\cdot u'}{u^2}\ ,\ \ C=const\\\\f'(x)=1+\dfrac{-2x}{x^4}=1-\dfrac{2}{x^3}\\\\f'(2)=1-\dfrac{2}{8}=1-0,25=0,75\\\\b)\ \ f(x)=\dfrac{x+1}{x^2}\ \ ,\ \ f(x)=\dfrac{1}{x}+\dfrac{1}{x^2}\\\\f'(x)=\dfrac{-1}{x^2}+\dfrac{-2x}{x^4}=-\dfrac{1}{x^2}-\dfrac{2}{x^3}\\\\f'(2)=-\dfrac{1}{4}-\dfrac{2}{8}=-\dfrac{4}{8}=-\dfrac{1}{2}$

Answer 2

1)f'(x)=(2/(2√x))*(x+1)+2√x=(1/√x)*(x+1)+2√x;

f'(4)=(1/√4)*(4+1)+2√4=2.5+4=6.5

2)f'(x)=x+(1/x²)=1-2/х³

f'(2)=1-0.25=0.75

Если второе условие такое, f(x)=(x+1)/x², то

f'(x)=((x+1)/x²)'=(x²-2x*(x+1))/x⁴=(-x²-2x)/x⁴

f'(2)=(-4-4)/16=-0.5

Answer 3

To find the derivative of a function, we can use the power rule, product rule, and chain rule as needed. Let's calculate the derivatives of the given functions:

1. Function: f(x) = 2√x * (x + 1)

Using the product rule, we can differentiate this function:

f'(x) = (2√x)' * (x + 1) + 2√x * (x + 1)'

To find the derivative of 2√x, we can apply the chain rule. Let's break it down step by step:

Let u = 2√x Then u' = (d/dx) 2√x

To find u', we can differentiate 2√x using the chain rule:

u' = (2√x)' = 2 * (1/2) * (x)^(-1/2) = x^(-1/2)

Now let's find the derivative of (x + 1):

(x + 1)' = 1

Substituting these results back into the product rule formula, we get:

f'(x) = x^(-1/2) * (x + 1) + 2√x * 1 = x^(-1/2) * (x + 1) + 2√x

To find f'(4), we substitute x = 4 into f'(x):

f'(4) = 4^(-1/2) * (4 + 1) + 2√4 = 2/√4 * 5 + 4 = 10/2 + 4 = 5 + 4 = 9

Therefore, f'(4) = 9.

2. Function: f(x) = (x + 1) / x^2

To find the derivative of this function, we can use the quotient rule:

f'(x) = [(x^2)' * (x + 1) - (x + 1)' * (x^2)] / (x^2)^2

Let's differentiate the terms using the power rule:

(x^2)' = 2x (x + 1)' = 1

Substituting these results into the quotient rule formula, we have:

f'(x) = [2x * (x + 1) - 1 * (x^2)] / (x^2)^2 = (2x^2 + 2x - x^2) / x^4 = (x^2 + 2x) / x^4 = x(x + 2) / x^4 = (x + 2) / x^3

To find f'(2), we substitute x = 2 into f'(x):

f'(2) = (2 + 2) / 2^3 = 4 / 8 = 1/2

Therefore, f'(2) = 1/2.