Find the Second Derivative f(x)=9x^3+1/x

Solution:

Step 1: Compute the first derivative of the function.

Step 1.1: Apply the Sum Rule for differentiation.

The derivative of $f(x) = 9x^3 + \frac{1}{x}$ is the sum of the derivatives of each term:

$$f'(x) = \frac{d}{dx}(9x^3) + \frac{d}{dx}\left(\frac{1}{x}\right).$$

Step 1.2: Differentiate $9x^3$.

Step 1.2.1: Recognize that 9 is a constant.

The derivative of $9x^3$ is $9$ times the derivative of $x^3$:

$$9 \cdot \frac{d}{dx}(x^3).$$

Step 1.2.2: Apply the Power Rule.

The Power Rule gives us:

$$9 \cdot (3x^{3-1}) = 27x^2.$$

Step 1.2.3: Simplify the expression.

The derivative of $9x^3$ is:

$$27x^2.$$

Step 1.3: Differentiate $\frac{1}{x}$.

Step 1.3.1: Rewrite the term.

Express $\frac{1}{x}$ as $x^{-1}$:

$$27x^2 + \frac{d}{dx}(x^{-1}).$$

Step 1.3.2: Apply the Power Rule.

Using the Power Rule, we find:

$$27x^2 - x^{-2}.$$

Step 1.4: Express the derivative using positive exponents.

The first derivative is:

$$f'(x) = 27x^2 - \frac{1}{x^2}.$$

Step 2: Calculate the second derivative of the function.

Step 2.1: Apply the Sum Rule for differentiation.

The second derivative is the sum of the derivatives of each term in $f'(x)$:

$$f''(x) = \frac{d}{dx}(27x^2) + \frac{d}{dx}\left(-\frac{1}{x^2}\right).$$

Step 2.2: Differentiate $27x^2$.

Step 2.2.1: Recognize that 27 is a constant.

The derivative of $27x^2$ is $27$ times the derivative of $x^2$:

$$27 \cdot \frac{d}{dx}(x^2).$$

Step 2.2.2: Apply the Power Rule.

The Power Rule gives us:

$$27 \cdot (2x^{2-1}) = 54x.$$

Step 2.2.3: Simplify the expression.

The derivative of $27x^2$ is:

$$54x.$$

Step 2.3: Differentiate $-\frac{1}{x^2}$.

Step 2.3.1: Apply the Power Rule.

Express $-\frac{1}{x^2}$ as $-x^{-2}$ and differentiate:

$$\frac{d}{dx}(-x^{-2}) = -(-2)x^{-2-1} = 2x^{-3}.$$

Step 2.3.2: Simplify the expression.

The derivative of $-\frac{1}{x^2}$ is:

$$2x^{-3}.$$

Step 2.4: Express the second derivative using positive exponents.

The second derivative is:

$$f''(x) = 54x + 2x^{-3} = 54x + \frac{2}{x^3}.$$

Step 3: Conclude with the second derivative.

The second derivative of $f(x)$ is:

$$f''(x) = 54x + \frac{2}{x^3}.$$

Knowledge Notes:

1. Sum Rule: The derivative of a sum of functions is the sum of the derivatives of each function.

2. Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

3. Constant Multiple Rule: If $c$ is a constant and $f(x)$ is a function, then the derivative of $cf(x)$ is $c$ times the derivative of $f(x)$.

4. Negative Exponent Rule: For any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$.

5. Chain Rule: If a function $y = f(u)$ and $u = g(x)$ are both differentiable, then $y$ is a differentiable function of $x$ and $dy/dx = (dy/du) \cdot (du/dx)$.

6. Product Rule: If $f(x)$ and $g(x)$ are both differentiable, then the derivative of their product $f(x)g(x)$ is $f'(x)g(x) + f(x)g'(x)$.

7. Differentiation of Constants: The derivative of a constant is zero.

8. Combining Exponents Rule: For any nonzero number $a$, $a^m \cdot a^n = a^{m+n}$.

These rules and properties are fundamental to calculus and are used extensively in differentiation to find the derivatives of various functions. The Power Rule and Sum Rule are particularly useful for polynomial functions, while the Chain Rule is essential for composite functions. The Constant Multiple Rule allows us to pull out constants from derivatives, simplifying the differentiation process. The Negative Exponent Rule helps in dealing with terms that have negative exponents, and the Product Rule is used when differentiating products of functions. The Differentiation of Constants and Combining Exponents Rule are also frequently used in simplifying expressions before or after taking derivatives.