Question

Find the derivative of the following functions. $$f(x)=3 x^{-9}$$

Short answer

Answer: The derivative of the function $$f(x)=3 x^{-9}$$ is $$f'(x)=-27x^{-10}$$.

Step-by-step solution

Rewrite the function

Rewrite the given function $$f(x)=3 x^{-9}$$ as follows: $$f(x)=3x^{-9}$$.

Apply the power rule

Next, apply the power rule to find the derivative of the function: $$f'(x)=-9(3)x^{-9-1}$$.

Simplify the expression

Now, simplify the expression for the derivative: $$f'(x)=-27x^{-10}$$.

Final answer

The derivative of the function $$f(x)=3 x^{-9}$$ is: $$f'(x)=-27x^{-10}$$.

Key concepts

Power Rule

When you encounter a function that involves a power of a variable, such as \(x^n\), the power rule is a simple and efficient way to find its derivative. The power rule states that if \(f(x) = x^n\), then its derivative \(f'(x)\) is \(n \cdot x^{n-1}\). This means you take the exponent, multiply the whole term by it, and then reduce the exponent by one.

Let's apply this to the original function \(f(x) = 3x^{-9}\):

• The exponent here is \(-9\).
• To find the derivative using the power rule, multiply \(-9\) by the coefficient \(3\), which gives you \(-27\).
• Then, reduce the exponent by one, resulting in a new exponent of \(-10\).

Now, you have the derivative in its preliminary form, \(f'(x) = -27x^{-10}\). This demonstrates how the power rule transforms seemingly complex expressions into manageable derivatives.

Simplification of Expressions

Once you've applied any differentiation rules, like the power rule, you often need to simplify the resulting expression to make it more manageable or to present it in its most understandable form.

In the example of finding the derivative of \(f(x) = 3x^{-9}\), after applying the power rule, we end up with \(-27x^{-10}\). Simplification usually involves:

• Ensuring all coefficients are combined properly, as done when \(-9 \times 3 = -27\).
• Rewriting any negative exponents, if necessary, into fractions, although in this exercise, it was unnecessary to rewrite \(x^{-10}\).

This step is essential as it ensures accuracy and clarity, enabling further mathematical operations to be performed on the expression if needed. Always double-check each simplification step to avoid errors, maintaining consistency in your results.

Differentiation

Differentiation is a fundamental process in calculus that involves finding how a function changes at any point. It’s essentially the mathematical way to describe the rate of change or slope of a function.

In our example, we needed to differentiate \(f(x) = 3x^{-9}\). The process:

• Begins by identifying which rules of differentiation to apply, such as the power rule for terms like \(x^n\).
• Involves meticulous application of these rules, ensuring the correct application of operations like multiplication and subtraction of exponents.
• Ends with a properly simplified expression that represents the derivative of the original function.

Differentiation allows us to analyze and understand the behavior of functions, making it a powerful tool in both mathematics and applied sciences. By mastering differentiation, you gain insights into the dynamic change of quantities in various contexts.