Differentiate the following functions using the appropriate notation: a) g(x) = 4/x^3 b) y = (x^3 - 5x)/x

Understand the Problem

The question is asking us to find the derivative of two functions a) g(x) = 4/x^3 and b) y = (x^3 - 5x)/x. For each function, we need to apply the appropriate differentiation rules and notation to find the derivative.

Answer

a) $g'(x) = -\frac{12}{x^4}$ b) $\frac{dy}{dx} = 2x$

Steps to Solve

Rewrite the function g(x)

Rewrite the function $g(x) = \frac{4}{x^3}$ using a negative exponent to make it easier to differentiate.

$g(x) = 4x^{-3}$

Differentiate g(x) using the power rule

Apply the power rule for differentiation, which states that if $f(x) = ax^n$, then $f'(x) = nax^{n-1}$.

$g'(x) = -3 \cdot 4x^{-3-1}$ $g'(x) = -12x^{-4}$

Rewrite g'(x) with a positive exponent

Rewrite the derivative with a positive exponent:

$g'(x) = -\frac{12}{x^4}$

Simplify the function y

Simplify $y = \frac{x^3 - 5x}{x}$ by dividing each term in the numerator by $x$:

$y = \frac{x^3}{x} - \frac{5x}{x}$ $y = x^2 - 5$

Differentiate y using the power rule

Differentiate $y = x^2 - 5$ with respect to $x$. The derivative of $x^2$ is $2x$, and the derivative of a constant (-5) is 0.

$\frac{dy}{dx} = 2x - 0$ $\frac{dy}{dx} = 2x$

a) $g'(x) = -\frac{12}{x^4}$ b) $\frac{dy}{dx} = 2x$

More Information

The power rule is a fundamental concept in differential calculus. It allows us to easily find the derivative of power functions which is shown in both parts of this question.

Tips

Forgetting to rewrite the original function with negative exponents before differentiating.
Incorrectly applying the power rule (e.g., not multiplying by the exponent or not subtracting 1 from the exponent).
Not simplifying the function before differentiating (part b), which can make the differentiation process more complex.
Forgetting that the derivative of a constant is zero.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!