Problem 21 - Calculus and its applications

Question

Find \(\frac{d y}{d x}\). $$ y=\frac{6}{x^{4}} $$

Step-by-Step Solution

Answer

\( \frac{d y}{d x} = \frac{-24}{x^5} \).

Step 1: Rewrite the Function

First, you rewrite the original function to enable us to use the power rule of differentiation. The given function is \( y = \frac{6}{x^4} \). We can rewrite this as \( y = 6x^{-4} \). This manipulation involves recognizing that \( x^{-n} = \frac{1}{x^n} \).

Step 2: Apply the Power Rule

Now differentiate using the power rule, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Apply this to \( y = 6x^{-4} \). Differentiating, we have \( \frac{d}{dx}(6x^{-4}) = 6 \cdot (-4)x^{-4-1} = -24x^{-5} \).

Step 3: Simplify the Result

Finally, simplify the derivative. The result is \( -24x^{-5} \). Alternatively, this can be written as \( \frac{-24}{x^5} \) to express it with positive exponents or as a fraction.

Key Concepts

Power Rule

The power rule is a fundamental tool in calculus for finding the derivative of expressions based on a variable raised to a power. If you have a function of the form \( x^n \), the power rule tells you its derivative is \( nx^{n-1} \). This makes differentiation straightforward by taking the exponent, multiplying it by the coefficient, and reducing the exponent by one.

So, if we start with \( y = 6x^{-4} \), we apply the power rule as follows: First, multiply the coefficient (6) by the exponent (-4), which gives -24. Then, subtract 1 from the exponent to get \( -5 \). The derivative is thus \( -24x^{-5} \).

This rule is particularly useful for its simplicity, allowing us to quickly find derivatives of polynomial functions as well as functions that involve negative or fractional exponents.

Derivative of a Function

The derivative of a function represents the rate at which the function's value changes as its input changes. It is a cornerstone concept in calculus. In essence, it measures how a function reacts to small changes in its input, which is a fancy way of saying how fast the output or the value of the function is changing at any point.

• The process of finding a derivative is known as differentiation.
• Derivatives are used to find slopes of tangents to curves, to calculate rates of change, and to solve problems in various fields like physics and engineering.

For the function \( y = \frac{6}{x^4} \) or equivalently \( y = 6x^{-4} \), its derivative \( \frac{d y}{d x} \) tells us how \( y \) changes with respect to \( x \). After applying differentiation here, through the power rule, we found the derivative to be \( -24x^{-5} \). This indicates how fast \( y \) decreases as \( x \) increases, given the negative sign of the derivative.

Manipulating Exponents

Manipulating exponents is an important skill when working with functions, particularly when preparing to apply calculus techniques like differentiation. When you see a fraction or a negative exponent, being able to rewrite it can make differentiation much more manageable.

Take, for instance, \( \frac{6}{x^4} \). This can instead be expressed as \( 6x^{-4} \), through the rule \( x^{-n} = \frac{1}{x^n} \). Understanding this allows us to convert division into multiplication, paving the way for using the power rule effectively.

By manipulating exponents in this way, you simplify complex expressions and turn them into forms that are much easier to differentiate or integrate.

• Negative exponents indicate the reciprocal of the base raised to the positive of that exponent.
• When applied correctly, manipulating exponents keeps expressions simpler and calculations more intuitive.

Mastering exponent manipulation is crucial for tackling a broad range of mathematical problems efficiently.