Problem 26 - Single Variable Calculus

Question

Find the derivative of the function. \(f(x)=\frac{5}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x}+200\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\frac{5}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x}+200\) is: \(f'(x)=\frac{-15}{x^4}+\frac{4}{x^3}+\frac{1}{x^2}\).

Step 1: Rewrite the function using exponent notation

To make finding the derivative easier, we can rewrite the function using exponent notation. The function becomes: \[f(x)=5x^{-3}-2x^{-2}-x^{-1}+200\]

Step 2: Apply the power rule to each term

Now, apply the power rule to find the derivatives of each term in the function: Derivative of \(5x^{-3}\): \[\frac{d}{dx}(5x^{-3})=(-3)(5)x^{-3-1}=-15x^{-4}\] Derivative of \(-2x^{-2}\): \[\frac{d}{dx}(-2x^{-2})=(-2)(-2)x^{-2-1}=4x^{-3}\] Derivative of \(-x^{-1}\): \[\frac{d}{dx}(-x^{-1})=(-1)(-1)x^{-1-1}=x^{-2}\] Derivative of \(200\): \[\frac{d}{dx}(200)=0\]

Step 3: Combine the derivatives

Now, combine the derivatives of each term to get the final result: \[f'(x)=-15x^{-4} + 4x^{-3} + x^{-2}\]

Step 4: Rewrite the derivatives using fractional notation (Optional)

If you want to write the final derivative in fractional notation, you can rewrite it as follows: \[f'(x)=\frac{-15}{x^4}+\frac{4}{x^3}+\frac{1}{x^2}\] So the derivative of the function \(f(x)=\frac{5}{x^{3}}-\frac{2}{x^{2}}-\frac{1}{x}+200\) is: \[f'(x)=\frac{-15}{x^4}+\frac{4}{x^3}+\frac{1}{x^2}\]

Key Concepts

Power Rule

Understanding the power rule for derivatives is crucial when studying calculus. It's a shortcut that makes finding derivatives of polynomials straightforward. The power rule states that if you have a function in the form of f(x) = xⁿ, where n is any real number, the derivative of this function f'(x) will be nx^n-1.

Applying this powerful tool means simply multiplying the power by the coefficient and then subtracting one from the power. This allows us to quickly find the derivatives of each term independently without complex calculations. It's especially useful for monomials, which are single-term expressions like 3x⁴ or -7x².

For example, to find the derivative of 3x⁴, you would multiply the exponent 4 by the coefficient 3, and then reduce the exponent by 1, resulting in 12x³. Remember that this applies no matter if the exponent is positive or negative, as seen in the original exercise.

Derivative of Polynomial

Polynomials are algebraic expressions consisting of multiple terms with variables raised to whole-number exponents, like x² or x⁵ - 3x³ + 2. When it comes to finding the derivative of a polynomial, the simplicity of the power rule shines. Since derivatives operate term-by-term, you can apply the power rule to each individual term of the polynomial separately.

In our textbook exercise, we see a polynomial with negative exponents. Even though they’re not typically seen in standard polynomial expressions, the power rule still applies universally. So, to find the derivative of any term in a polynomial expression, regardless of whether the exponent is positive or negative, you multiply the exponent by the coefficient of the term and decrease the exponent by one.

Deriving each term in the original exercise independently and then combining the results give us the derivative of the entire polynomial, ensuring no term is overlooked. The derivative of a constant, such as 200, is zero, as it doesn't change regardless of the value of x.

Exponent Notation

Exponent notation is a way of expressing powers or repeated multiplication of a number by itself. It is extremely useful in calculus, particularly when dealing with derivatives. In exponent notation, xⁿ represents x multiplied by itself n times. When n is a negative integer, it signifies division by the number x raised to the positive value of n, for example, x^-3 is the same as 1/x³.

In calculus, using exponent notation is often more convenient than working with roots or fractions because it simplifies the application of differentiation rules, like the power rule mentioned earlier. Rewriting a function in terms of exponent notation, as seen in the step-by-step solution for our exercise, makes it easier to direct apply the power rule to find derivatives, particularly when you encounter terms with negative exponents or complex polynomial expressions.