Question

Derivatives Find and simplify the derivative of the following functions. $$y=\frac{4}{p^{3}}$$

Short answer

Answer: The derivative of the function $$y=\frac{4}{p^3}$$ with respect to $$p$$ is $$\frac{dy}{dp}=\frac{-12}{p^4}$$.

Step-by-step solution

Rewrite the equation

To make the differentiation process easier, we can rewrite the given function: $$y=\frac{4}{p^{3}} = 4p^{-3} $$

Apply the power rule

Now we need to apply the power rule of differentiation, which states that $$\frac{d}{dp}(p^n) = n \cdot p^{n-1}$$ In this case, we have $$n=-3$$, so applying the power rule, we get: $$\frac{dy}{dp}=-3(4)p^{-3-1} $$

Simplify the expression

Now, simplify the expression by combining the constants and the powers of $$p$$: $$\frac{dy}{dp}=-12p^{-4} $$

Rewrite in the original form

Finally, let's rewrite the expression in the same form as the original function: $$\frac{dy}{dp}=\frac{-12}{p^4}$$ So, the derivative of the function $$y=\frac{4}{p^3}$$ with respect to $$p$$ is $$\frac{dy}{dp}=\frac{-12}{p^4}$$.

Key concepts

Power Rule

The power rule is a fundamental technique in calculus, especially when dealing with derivatives of power functions. Its simplicity allows us to differentiate terms of the form \( p^n \) directly. The rule states that for a term \( p^n \), where \( n \) is any real number, the derivative \( \frac{d}{dp}(p^n) \) is \( n \times p^{n-1} \). Simply put, you multiply the exponent \( n \) by the coefficient of the term, then decrease the exponent by one.

Consider an example where \( y = x^5 \). Applying the power rule, the derivative would be \( 5 \times x^{4} = 5x^4 \). This straightforward application is what makes the power rule so beneficial. It transforms the differentiation of even complex powers into a rote process. Understanding and practicing the power rule is crucial, as it is often used in combination with other differentiation rules in calculus.

When you encounter terms like \( \frac{4}{p^3} \), rewriting them as \( 4p^{-3} \) (transforming the negative exponent) can make the application of the power rule easier. This simplification helps in consistent application across different problems.

Derivative of Power Functions

Power functions entail expressions where a variable is raised to a power, such as \( p^n \). To find the derivative of any power function, one can effectively utilize the power rule, as we discussed earlier.

In calculus, the process of finding the derivative of power functions is key, as it offers insights into the behavior of a function's rate of change. When facing a problem where the function is presented in a fraction form, it's beneficial to convert it into a format compatible with the power rule. This often involves expressing the denominator with a negative exponent, making it easier to differentiate directly.

Take for instance \( y = \frac{4}{p^3} \), converting it to \( y = 4p^{-3} \) simplifies the differentiation process.

• Apply the power rule: \( \frac{d}{dp}(4p^{-3}) = -3 \times 4p^{-4} \). This calculation reveals both the rate at which the function changes and positions us to simplify further.

These transformations not only streamline the computational effort but also enhance our understanding of the underlying mathematical behavior of such functions.

Simplifying Derivatives

Simplifying derivatives is a crucial step that often follows the differentiation process. After applying the differentiation rules like the power rule, we arrive at a new expression which needs to be put into a more understandable or simplified form.

Let's revisit our example: after differentiating \( 4p^{-3} \) to \( -12p^{-4} \), the simplification clarifies how the function inherently behaves. Here, the coefficient \(-12\), derived from multiplying constants and exponents, is maintained and the power of \( p \) expressed positively: \( \frac{-12}{p^4} \).

• The negative exponent \(-4\) is translated back into a fraction, aligning with the original problem's format.
• Remember, the aim is to simplify the expression to match the most common form, making interpretations easy and insightful.

Simplification doesn't merely involve reducing the expression - it carries significance in comprehending any additional changes to the function over its domain. Hence, mastering simplification practices can greatly enhance your mathematical fluency and solving efficiency in calculus.