Explanation

1. Leibniz’s theorem gives a formula to find the nth derivative of the product of two functions.

2. It uses the binomial expansion:

Explanation

 We are given the equation:

30y=60

To find the value of y, divide both sides by 30:

𝑦=60/30=2

Explanation

We know that the difference of their squares is 84, so we can write the equation:

(x+2)^2 - x^2 = 84

Expanding the equation, we get:

x^2 + 4x + 4 - x^2 = 84

Simplifying the equation, we get:

4x + 4 = 84

Subtracting 4 from both sides, we get:

4x = 80

Dividing both sides by 4, we get:

x = 20

So, the two consecutive even numbers are 20 and 22.

The sum of these numbers is:

20 + 22 = 42

Explanation

If $a + 3b = 6$ and $ab=3$, then the value of $a^2 + 9b^2$ is:

We are given:

a+3b=6(1),ab=3(2)

$$$$a+3b=6(1),ab=3(2)

Now square equation (1):

$$(a + 3b)^2 = 6^2 = 36 Rightarrow a^2 + 9b^2 + 6ab = 36$$

Use $ab = 3$

$$a^2 + 9b^2 + 6(3) = 36 Rightarrow a^2 + 9b^2 + 18 = 36 Rightarrow a^2 + 9b^2 = 18$$

Explanation

u . n = 0, where n is the unit normal to the boundary

This is the impermeability condition, which states that the fluid cannot flow through the fixed boundary.

Explanation

To evaluate the limit:

lim (x→0) (√(x^2+100) - 10)/x^2

Rationalize the numerator:

= lim (x→0) ((x^2+100) - 100) / (x^2(√(x^2+100) + 10))

= lim (x→0) x^2 / (x^2(√(x^2+100) + 10))

Cancel x^2:

= lim (x→0) 1 / (√(x^2+100) + 10)

As x→0, √(x^2+100) → √100 = 10:

= 1 / (10 + 10)

= 1 / 20

Explanation

Explanation

To find the value of y in the equation 3x+y=2, we can substitute the given value of x into the equation and solve for y.

1. Given value of x: x = 4.

2. Substitute the value of x into the equation: 3(4) + y = 2.

3. Simplify the equation: 12 + y = 2.

4. Solve for y by isolating the variable: y = 2 - 12.

5. Simplify the right side of the equation: y = -10.

Therefore, when the value of x is 4 in the equation 3x+y=2, the value of y is -10.

Explanation

The identity element e should satisfy ae = ea = a for all a in G.

Here, e = 6 works since 26=2, 46=4, 66=6, 86=8.

Explanation

Given y = 3:

Step 1: Calculate y³.

y³ = 3³ = 27

Step 2: Calculate y³ - y.

y³ - y = 27 - 3 = 24

Step 3: Calculate y³(y³ - y).

y³(y³ - y) = 27 × 24

= 648