Find the derivative of 1/√(x^4)
Understand the Problem
The question is asking to find the derivative of the function 1/√(x^4). We will need to apply the rules of differentiation to solve this problem step by step.
Answer
The derivative is $-\frac{2}{x^3}$.
Answer for screen readers
The derivative of the function $\frac{1}{\sqrt{x^4}}$ is $-\frac{2}{x^3}$.
Steps to Solve
- Rewrite the function in a simpler form
To make differentiation easier, rewrite the function $\frac{1}{\sqrt{x^4}}$ in exponent form. We know that $\sqrt{x^4} = x^{4/2} = x^{2}$. Therefore, our function can be expressed as:
$$ f(x) = x^{-2} $$
- Differentiate the function
Now, apply the power rule for differentiation, which states that if $f(x) = x^n$, then $f'(x) = n \cdot x^{n-1}$. Here, $n = -2$:
$$ f'(x) = -2 \cdot x^{-2-1} = -2 \cdot x^{-3} $$
- Rewrite the derivative in standard form
Finally, we can rewrite the derivative back into a simpler form. Since $x^{-3} = \frac{1}{x^3}$, our derivative becomes:
$$ f'(x) = -\frac{2}{x^3} $$
The derivative of the function $\frac{1}{\sqrt{x^4}}$ is $-\frac{2}{x^3}$.
More Information
The derivative gives the rate of change of the function at any point. In this case, $-\frac{2}{x^3}$ indicates that the function decreases as $x$ increases. The negative sign shows that the function is decreasing throughout its domain.
Tips
- Forgetting to apply the negative exponent when differentiating.
- Not rewriting the function in exponent form, which can complicate differentiation.
- Incorrectly simplifying the final derivative.