Differentiate the following functions using appropriate notation: a) g(x) = 4/x^3 b) y = (x^3 - 5x)/x
Understand the Problem
The question asks us to differentiate two functions using appropriate notation. The functions are a) g(x) = 4/x^3 and b) y = (x^3 - 5x)/x. We need to apply differentiation rules to find the derivatives of these functions.
Answer
a) $g'(x) = \frac{-12}{x^4}$ b) $\frac{dy}{dx} = 2x$
Answer for screen readers
a) $g'(x) = \frac{-12}{x^4}$ b) $\frac{dy}{dx} = 2x$
Steps to Solve
- Differentiate $g(x) = \frac{4}{x^3}$
First rewrite $g(x)$ using a negative exponent: $g(x) = 4x^{-3}$ Then, apply the power rule: $\frac{d}{dx}(ax^n) = nax^{n-1}$ $g'(x) = -3 \cdot 4x^{-3-1}$ $g'(x) = -12x^{-4}$ Finally, rewrite the derivative with a positive exponent: $g'(x) = \frac{-12}{x^4}$.
- Differentiate $y = \frac{x^3-5x}{x}$
Simplify the expression for $y$ by dividing each term in the numerator by $x$: $y = \frac{x^3}{x} - \frac{5x}{x}$ $y = x^2 - 5$ Now, differentiate $y$ with respect to $x$, applying the power rule and the constant rule: $\frac{dy}{dx} = \frac{d}{dx}(x^2) - \frac{d}{dx}(5)$ $\frac{dy}{dx} = 2x - 0$ $\frac{dy}{dx} = 2x$
a) $g'(x) = \frac{-12}{x^4}$ b) $\frac{dy}{dx} = 2x$
More Information
The power rule is a fundamental rule in calculus used to differentiate functions of the form $x^n$, where $n$ is a real number. The constant rule states that the derivative of a constant is zero.
Tips
A common mistake is not simplifying the expression of $y$ before differentiating, which would require using the quotient rule and is more complex. Another common mistake is forgetting to apply the chain rule when it is required (although it is not needed in these problems). Finally, mistakes can be made when applying the power rule, especially with negative exponents.
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