Find the Second Derivative f(x)=4/x
The problem is asking for the calculation of the second derivative of the given function, which is a mathematical procedure to find the rate at which the slope of the tangent to the function's curve is changing. The function in question is f(x) = 4/x, and you are being asked to differentiate it twice with respect to x to find the second derivative, often denoted as f''(x) or d²/dx². This will involve applying the rules of differentiation to the given function.
$f \left(\right. x \left.\right) = \frac{4}{x}$
Answer
Solution:
Step 1: Compute the first derivative of $f(x)$.
1.1. The derivative of $f(x) = \frac{4}{x}$ is found by differentiating the function with respect to $x$. Apply the constant multiple rule to $4 \cdot \left(\frac{d}{dx}x^{-1}\right)$.
1.2. Express $\frac{1}{x}$ as $x^{-1}$ to simplify differentiation.
1.3. Apply the power rule for differentiation, which gives the derivative of $x^n$ as $nx^{n-1}$, where $n$ is a real number. In this case, $n = -1$.
1.4. Calculate the derivative: $4 \cdot (-1)x^{-2}$.
1.5. Simplify the expression.
1.5.1. Use the rule for negative exponents, which states that $x^{-n} = \frac{1}{x^n}$.
1.5.2. Combine the constant and the term with the exponent.
1.5.2.1. Combine $-4$ with $\frac{1}{x^2}$ to get $\frac{-4}{x^2}$.
1.5.2.2. Place the negative sign in front of the fraction to obtain the first derivative, $f'(x) = -\frac{4}{x^2}$.
Step 2: Determine the second derivative of $f(x)$.
2.1. The derivative of $f'(x) = -\frac{4}{x^2}$ is found by differentiating the function with respect to $x$. Apply the constant multiple rule to $-4 \cdot \left(\frac{d}{dx}x^{-2}\right)$.
2.2. Utilize the rules for exponents.
2.2.1. Express $\frac{1}{x^2}$ as $(x^2)^{-1}$.
2.2.2. Apply the rule for exponents when raising a power to a power, which is $(a^m)^n = a^{mn}$.
2.2.2.1. Apply the power rule for differentiation.
2.2.2.2. Multiply the exponent $2$ by $-1$ to get $x^{-2}$.
2.3. Differentiate using the power rule, where the derivative of $x^n$ is $nx^{n-1}$, with $n = -2$ in this case.
2.4. Perform the multiplication: $-2 \cdot -4 = 8$, and then apply it to the term $x^{-3}$.
2.5. Simplify the expression.
2.5.1. Rewrite the expression using the negative exponent rule.
2.5.2. Combine the constant $8$ with the term $\frac{1}{x^3}$ to obtain the second derivative, $f''(x) = \frac{8}{x^3}$.
Step 3: Conclude the second derivative.
The second derivative of $f(x)$ with respect to $x$ is $f''(x) = \frac{8}{x^3}$.
Knowledge Notes:
Constant Multiple Rule: When taking the derivative of a constant multiplied by a function, the derivative is the constant multiplied by the derivative of the function.
Power Rule: A fundamental rule of differentiation that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
Negative Exponent Rule: For any nonzero number $b$ and any integer $n$, $b^{-n} = \frac{1}{b^n}$.
Combining Terms: When simplifying expressions, similar terms are combined to form a single term, often resulting in a more compact expression.
Differentiation of Rational Functions: When differentiating functions of the form $\frac{1}{x^n}$, it is often helpful to rewrite them as $x^{-n}$ and then apply the power rule.
