Question

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

Short answer

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

Step-by-step solution

Identify the function to be derived

The given function is $$f(x) = 3x^{-9}$$. We need to find its derivative with respect to x, $$\frac{df(x)}{dx}$$ or $$f'(x)$$.

Apply the constant rule

Since 3 is a constant, we can move it out of the derivative. The constant rule states that if $$f(x) = c * g(x)$$, then $$f'(x) = c * g'(x)$$ where c is a constant and g(x) is a function of x. In this case, $$c = 3$$ and $$g(x) = x^{-9}$$. Therefore, we can write: $$f'(x) = 3 * g'(x)$$

Find the derivative of g(x)

Now, we need to find the derivative of g(x) with respect to x. Our function g(x) is $$x^{-9}$$. We will use the power rule, which states that if $$g(x) = x^n$$ then $$g'(x) = nx^{n-1}$$. In our case, $$n = -9$$, so the derivative of g(x) is: $$g'(x) = -9x^{-9-1}$$ Simplifying the exponent, we get: $$g'(x) = -9x^{-10}$$

Substitute g'(x) into the expression for f'(x)

Now we will substitute the information from step 3 back into the expression we found in step 2: $$f'(x) = 3 * g'(x)$$ Substituting $$g'(x) = -9x^{-10}$$, we get: $$f'(x) = 3 * (-9x^{-10})$$

Simplify the expression

Now, simply multiply the constants together and write the simplified expression for f'(x): $$f'(x) = -27x^{-10}$$ The derivative of the function $$f(x) = 3x^{-9}$$ is $$f'(x) = -27x^{-10}$$

Key concepts

Constant Rule

In calculus, the constant rule is an essential tool to simplify the process of differentiation. When differentiating a function, constants can seem a bit tricky, but the constant rule helps to clarify things.

The rule states that when you have a constant multiplied by a function, such as in the general form: \(f(x) = c \cdot g(x)\), the derivative of \(f(x)\) (denoted as \(f'(x)\)) can be found by multiplying the constant \(c\) with the derivative of \(g(x)\). Simply put:

• If \(c\) is a constant, then \(\frac{d}{dx}[c \cdot g(x)] = c \cdot \frac{d}{dx}[g(x)]\).

For the given function \(f(x) = 3x^{-9}\), the constant rule allows us to take the constant \(3\) out of the differentiation process. Therefore, the differentiation focuses entirely on the power of \(x\), i.e., \(x^{-9}\).

This simplifies the derivative into two manageable parts: a constant, and another expression whose derivative we need to determine.

Power Rule

The power rule is another fundamental concept in calculus used to easily find the derivative of functions with powers of \(x\). If you have a function in the form \(g(x) = x^n\), its derivative is determined by bringing the exponent \(n\) down to multiply the function, and then subtracting 1 from the exponent. This can be expressed as:

• \(\frac{d}{dx}[x^n] = n \cdot x^{n-1}\)

In our function \(f(x) = 3x^{-9}\), the expression \(x^{-9}\) fits this form perfectly. Using the power rule, the derivative of \(x^{-9}\) becomes \(-9x^{-10}\). Here, the exponent \(-9\) is moved to the front, and then it is decreased by 1, forming the new expression:

\(g'(x) = -9x^{-10}\).

The power rule is particularly useful when dealing with polynomials or functions that involve powers of variables, making it simpler to calculate derivatives efficiently.

Calculus

Calculus serves as the mathematical study of change and motion, with differentiation being one of its core methods. Through differentiation, we can determine the rate of change of a quantity. This has applications in physics, engineering, economics, and everyday problem-solving.

Calculus splits into two main branches:

• Differential calculus, focusing on derivatives and rates of change.
• Integral calculus, which deals with integrals and accumulations of quantities.

Understanding basic derivative rules, such as the constant and power rules, forms the foundation for exploring more advanced concepts in calculus. This process starts with simple functions and expands to complex real-world systems. It empowers us to analyze how functions behave, transform, and influence the environments around us.

In practical terms, mastering calculus and its principles allows for precise modeling and prediction of dynamic systems, offering critical insight into both theoretical and applied sciences.