Higher Order Derivatives: Examples | nool - Free Printable

Problem:

Find the third derivative of the function \( f(x) = x^5 + 2x^3 - x + 4 \).

Solution:

To find the third derivative of \( f(x) \), we need to compute the first, second, and then the third derivatives step by step. Let's proceed systematically.

#### Step 1: Compute the first derivative \( f'(x) \)
The given function is:
\[
f(x) = x^5 + 2x^3 - x + 4
\]

Using the power rule for differentiation, \( \frac{d}{dx} x^n = n x^{n-1} \), we differentiate each term:

1. The derivative of \( x^5 \) is \( 5x^4 \).
2. The derivative of \( 2x^3 \) is \( 2 \cdot 3x^2 = 6x^2 \).
3. The derivative of \( -x \) is \( -1 \).
4. The derivative of the constant \( 4 \) is \( 0 \).

Thus, the first derivative is:
\[
f'(x) = 5x^4 + 6x^2 - 1
\]

#### Step 2: Compute the second derivative \( f''(x) \)
Now, we differentiate \( f'(x) = 5x^4 + 6x^2 - 1 \):

1. The derivative of \( 5x^4 \) is \( 5 \cdot 4x^3 = 20x^3 \).
2. The derivative of \( 6x^2 \) is \( 6 \cdot 2x = 12x \).
3. The derivative of the constant \( -1 \) is \( 0 \).

Thus, the second derivative is:
\[
f''(x) = 20x^3 + 12x
\]

#### Step 3: Compute the third derivative \( f'''(x) \)
Now, we differentiate \( f''(x) = 20x^3 + 12x \):

1. The derivative of \( 20x^3 \) is \( 20 \cdot 3x^2 = 60x^2 \).
2. The derivative of \( 12x \) is \( 12 \cdot 1 = 12 \).

Thus, the third derivative is:
\[
f'''(x) = 60x^2 + 12
\]

Final Answer:

\[
\boxed{60x^2 + 12}
\]