Problem 12 - Brief Applied Calculus

Question

Find the derivative of each function. $$ h(x)=\frac{4}{x^{3}} $$

Step-by-Step Solution

Answer

The derivative of \( h(x) = \frac{4}{x^3} \) is \(-\frac{12}{x^4}\).

Step 1: Transform Function

To find the derivative of the function \( h(x) = \frac{4}{x^3} \), we first rewrite it in a more differentiable form. We can express \( \frac{4}{x^3} \) as \( 4x^{-3} \). This step uses the property that \( \frac{1}{x^n} = x^{-n} \).

Step 2: Differentiate Using Power Rule

Now that we have rewritten the function, \( h(x) = 4x^{-3} \), we can differentiate it using the power rule. The power rule states that \( \frac{d}{dx} x^n = nx^{n-1} \). Applying the power rule, we differentiate: \( \frac{d}{dx} [4x^{-3}] = 4 \cdot (-3) x^{-3-1} = -12x^{-4} \).

Step 3: Simplify Expression

The expression \(-12x^{-4}\) can be rewritten in a more standard form as \(-\frac{12}{x^4}\). The negative exponent indicates the reciprocal of the power of \(x\). This gives us the derivative in a more conventional form.

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that simplifies the process of finding derivatives for expressions involving powers of a variable. It is particularly useful because it provides a quick method for differentiating polynomial terms. The rule states that if you have a function of the form \( f(x) = x^n \), the derivative is given by \( f'(x) = nx^{n-1} \).
For example, when differentiating \( 4x^{-3} \), we use the power rule by multiplying the exponent, \(-3\), by the coefficient, \(4\), resulting in \(-12\). We then decrease the power of \(x\) by one, which changes it from \(-3\) to \(-4\).
Using the power rule effectively allows us to transform complex algebraic expressions into simpler derivatives, making calculus more approachable. It's a vital technique when dealing with polynomial expressions in calculus exercises.

Negative Exponents

Negative exponents are a mathematical notation that shows a reciprocal relationship. When an expression involves a negative exponent, it implies the reciprocal of that power.
It is expressed as \( x^{-n} = \frac{1}{x^n} \). This transformation is critical for simplifying expressions and performing derivative operations. It converts division into multiplication by a power of the base number \(x\).
In our function \( h(x) = \frac{4}{x^3} \), by rewriting it as \( 4x^{-3} \), we convert the divisor into a multiplication form, simplifying the differentiation process. This transformation harnesses negative exponents for easier manipulation and can be very useful in calculus and algebra.

Differentiation Steps

Differentiation involves a systematic process to find the derivative of a function.

• First, reformulate the function to a differentiable form, if necessary. For instance, rewrite any fractions with negative exponents.
• Next, apply the appropriate differentiation rule, such as the power rule, to find the derivative of each term in the expression.
• Finally, simplify the derivative expression. For example, rewrite terms with negative exponents into their fractional form if needed.

Following these steps ensures accuracy and consistency in finding derivatives. It makes complex functions more manageable and comprehensible, allowing for a clearer view of the underlying behavior of the original function.