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 $-\frac{2}{x^3}$.

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.

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