arignment following Denive the for 1) $$ \Delta x(10) $$ 2) $$ \Delta_{x}(5 x) $$ 3) $$ \Delta_{x}\left(10 x^{4}\right) $$ 4) $$ \Delta x\left(4 x^{3}+2 x^{2}+3 x+5\right) $$ 5) $$ D_{x}\left(10 x^{5}+4 x^{4}+2 x^{3}+4 x^{2}+10 x\right. $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's go through each of the problems step by step. It appears you're working with two different operators:

1. **Finite Difference Operator ($$ \Delta $$)**:
   $$
   \Delta f(x) = f(x + 1) - f(x)
   $$
   
2. **Differential Operator ($$ D_x $$)**:
   $$
   D_x f(x) = \frac{df(x)}{dx}
   $$

Let's tackle each problem accordingly.

---

### 1) $$ \Delta x(10) $$

**Interpretation**: Apply the finite difference operator to the constant function $$ f(x) = 10 $$.

**Solution**:
$$
\Delta f(x) = f(x + 1) - f(x) = 10 - 10 = 0
$$

**Answer**:
$$
\Delta x(10) = 0
$$

---

### 2) $$ \Delta_{x}(5x) $$

**Interpretation**: Apply the finite difference operator to $$ f(x) = 5x $$.

**Solution**:
$$
\Delta f(x) = f(x + 1) - f(x) = 5(x + 1) - 5x = 5x + 5 - 5x = 5
$$

**Answer**:
$$
\Delta_{x}(5x) = 5
$$

---

### 3) $$ \Delta_{x}\left(10x^{4}\right) $$

**Interpretation**: Apply the finite difference operator to $$ f(x) = 10x^4 $$.

**Solution**:
$$
\Delta f(x) = f(x + 1) - f(x) = 10(x + 1)^4 - 10x^4
$$
Expand $$ (x + 1)^4 $$:
$$
(x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1
$$
Thus,
$$
\Delta f(x) = 10(x^4 + 4x^3 + 6x^2 + 4x + 1) - 10x^4 = 40x^3 + 60x^2 + 40x + 10
$$

**Answer**:
$$
\Delta_{x}\left(10x^{4}\right) = 40x^3 + 60x^2 + 40x + 10
$$

---

### 4) $$ \Delta x\left(4x^{3} + 2x^{2} + 3x + 5\right) $$

**Interpretation**: Apply the finite difference operator to $$ f(x) = 4x^3 + 2x^2 + 3x + 5 $$.

**Solution**:
$$
\Delta f(x) = f(x + 1) - f(x)
$$
First, compute $$ f(x + 1) $$:
$$
f(x + 1) = 4(x + 1)^3 + 2(x + 1)^2 + 3(x + 1) + 5
$$
Expand each term:
$$
4(x + 1)^3 = 4(x^3 + 3x^2 + 3x + 1) = 4x^3 + 12x^2 + 12x + 4
$$
$$
2(x + 1)^2 = 2(x^2 + 2x + 1) = 2x^2 + 4x + 2
$$
$$
3(x + 1) = 3x + 3
$$
Combine all terms:
$$
f(x + 1) = 4x^3 + 12x^2 + 12x + 4 + 2x^2 + 4x + 2 + 3x + 3 + 5 = 4x^3 + 14x^2 + 19x + 14
$$
Now, subtract $$ f(x) $$:
$$
\Delta f(x) = (4x^3 + 14x^2 + 19x + 14) - (4x^3 + 2x^2 + 3x + 5) = 12x^2 + 16x + 9
$$

**Answer**:
$$
\Delta x\left(4x^{3} + 2x^{2} + 3x + 5\right) = 12x^2 + 16x + 9
$$

---

### 5) $$ D_{x}\left(10x^{5} + 4x^{4} + 2x^{3} + 4x^{2} + 10x\right) $$

**Interpretation**: Apply the differential operator to $$ f(x) = 10x^5 + 4x^4 + 2x^3 + 4x^2 + 10x $$.

**Solution**:
Differentiate each term with respect to $$ x $$:
$$
\frac{d}{dx}(10x^5) = 50x^4
$$
$$
\frac{d}{dx}(4x^4) = 16x^3
$$
$$
\frac{d}{dx}(2x^3) = 6x^2
$$
$$
\frac{d}{dx}(4x^2) = 8x
$$
$$
\frac{d}{dx}(10x) = 10
$$
Combine all derivatives:
$$
D_{x}\left(10x^{5} + 4x^{4} + 2x^{3} + 4x^{2} + 10x\right) = 50x^4 + 16x^3 + 6x^2 + 8x + 10
$$

**Answer**:
$$
D_{x}\left(10x^{5} + 4x^{4} + 2x^{3} + 4x^{2} + 10x\right) = 50x^4 + 16x^3 + 6x^2 + 8x + 10
$$

---

Feel free to ask if you need further clarification on any of these steps!
Here are the simplified answers for each problem: 1. $$ \Delta x(10) = 0 $$ 2. $$ \Delta_{x}(5x) = 5 $$ 3. $$ \Delta_{x}\left(10x^{4}\right) = 40x^3 + 60x^2 + 40x + 10 $$ 4. $$ \Delta x\left(4x^{3} + 2x^{2} + 3x + 5\right) = 12x^2 + 16x + 9 $$ 5. $$ D_{x}\left(10x^{5} + 4x^{4} + 2x^{3} + 4x^{2} + 10x\right) = 50x^4 + 16x^3 + 6x^2 + 8x + 10 $$

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