Derivative Check

Breaking down the function

First, I’m expressing $f(x)$ as $[x + \csc^4(x^9 -1)]^{-1}$. Then, I find the derivative of $[x + \csc^4(x^9 -1)]'$. Now, I’m on the next steps to proceed with the calculations.

Refining the derivative

I’m working through the derivative of $\csc^4(u)$ by expressing $u = x^9 -1$, which simplifies the expression to $4 \csc^3(u) \cdot \csc(u) \cot(u) \cdot u'$.

Breaking down the derivative

I’m taking a closer look at the derivative of $\csc^4(u)$. Using the chain rule, I find that $\frac{d}{dx} \csc^4(u) = -36 x^8 \csc^4(u) \cot(u)$.

Revisiting the derivative

I’m rethinking the derivative of $\csc^4(u)$ to ensure its accuracy and sign conventions are spot on.

Simplifying derivatives

I’m examining the derivative of $y = \csc^4(u)$ with respect to $x$, incorporating $\frac{du}{dx} = 9 x^8$. Progressing through each step, I’m comparing terms to align the derivative with the given expression.

Reassessing the term

I noticed a mismatch in the denominator between $[x + \csc^4(u)]^{2}$ and $x + [\csc(u)]^5$. This discrepancy might suggest $[x + \csc^4(u)]^{2}$ is equivalent to $[x + \csc^5(u)] \cdot [x + \csc^4(u)]$.

Confirming simplification

I’m noting progress on simplifying the derivative. The numerator now reads $-1 + 36 x^8 \csc^4(u) \cot(u)$, which intriguingly matches $-4 + 36 x^8 \csc(u) \cot(u)$ uniquely by examining the powers of $\csc(u)$.

Simplifying expressions

I’m thinking through potential simplifications of the expressions using $\csc^4(u)$ and $\cot(u)$ by factoring or rewriting.

Breaking down the calculation

I set $D = x + \csc^4(u)$ to simplify the derivative computation of $f(x)$. Ultimately, the obtained derivative, $f'(x)$, diverged from the user's initial expression.

No, the expression you provided is not the correct derivative of $f(x) = \frac{1}{x + \csc^4(x^9 - 1)}$. Let's compute the derivative step by step to find the correct expression.

Step 1: Define the function and its components

Let:

• $u = x^9 - 1$
• $f(x) = [x + \csc^4(u)]^{-1}$

Step 2: Differentiate $f(x)$ using the chain rule

The derivative of $f(x)$ is:

Step 3: Compute $\frac{d}{dx}[\csc^4(u)]$

First, find the derivative of $\csc^4(u)$ with respect to $u$:

Next, find $\frac{du}{dx}$:

Now, compute $\frac{d}{dx}[\csc^4(u)]$:

Step 4: Substitute back into the derivative of $f(x)$

Simplify the numerator:

Step 5: Final expression for $f'(x)$

Conclusion:

The correct derivative is:

This differs from the expression you provided, which has different powers of the cosecant function and a different denominator. Therefore, your expression is not the correct derivative of the given function.

Answer:

No; the correct derivative is different—the expression you provided is not the right answer.

OK, let me see. The user provided the expression for the derivative but it seems to be incorrect. The assistant then correctly computes and compares the derivative, determining the error lies in the original expression.

Identifying inconsistencies

I’m mapping out the discrepancies between the user's and assistant's expressions, focusing on differences in the numerator and denominator, and considering potential errors in differentiation or simplification.

Recomputing the derivative of f(x) = 1 / [x + csc^4(u)] with u = x^9 -1 and analyzing each step.

Taking a closer look

I’m simplifying the derivative of f(x) by using the chain rule to tackle (csc^4(u))' and substituting necessary terms. This results in a more concise form.

Pinpointing possible errors

I’m analyzing the problem to find if the user mistakenly applied the derivative to (csc^4(u)), confused the powers, or misused the chain rule.

Identifying an error

I noticed that the user's expression missed the fourth power of csc, which should be $\csc^4(u)$. The exponent of csc was incorrectly $1$.

Revisiting derivatives

I’m correcting the calculation of the derivative of csc^4(u) by noting csc'(u) is -csc(u) cot(u), leading to the corrected result: -36 x^8 csc^5(u) cot(u).

Identifying miscalculations

I’m gathering info on potential misapplications of the chain rule and derivative calculations involving csc^4(u), which might explain the incorrect derivative -4 csc^5(u) cot(u) u'.

Taking a closer look

I'm analyzing incorrect derivatives of csc(u). Options include -csc(u) cot(u), -csc^2(u) cot(u), and -csc(u) csc(u) cot(u). These aren't standard, and -csc^n(u) is also inaccurate.

Spotting the inconsistency

I noticed that the derivative of csc^4(u) should include a leading coefficient of -4, but it was missing in the user's expression.

Identifying the issue

I’m focusing on the user's calculation, potentially due to a misplaced constant term or an incorrect application of the derivative rules. This incorrect application might involve the numerator or the chain rule.

Identifying the mistake

OK, let me see. It seems the user miscalculated the derivative of $\csc^4(u)$ by mistaking it for $\csc(u)$, leading to missing powers of $\csc(u)$ in the derivative.

Certainly! The incorrect answer likely stems from a miscalculation in the differentiation of the term $\csc^4(x^9 - 1)$. Specifically, the error probably occurred in applying the chain rule and the power rule during differentiation. Let's identify the mistake step by step.

Step 1: Correct Differentiation of $\csc^4(u)$

First, let's recall that:

• The derivative of $\csc(u)$ with respect to $u$ is $-\csc(u)\cot(u)$.
• The derivative of $[f(u)]^n$ with respect to $u$ is $n[f(u)]^{n-1}f'(u)$.

So, the derivative of $\csc^4(u)$ with respect to $u$ is:

Step 2: Including the Chain Rule with $u = x^9 - 1$

Next, we find $\frac{du}{dx} = 9x^8$. So, the derivative of $\csc^4(u)$ with respect to $x$ is:

Step 3: Likely Error in the User's Calculation

It appears the user may have mistakenly calculated the derivative of $\csc^4(u)$ as:

Here, the error is omitting the exponent on $\csc(u)$ in the derivative. The user seems to have treated $\csc^4(u)$ as if it were $\csc(u)$ during differentiation, neglecting the $\csc^3(u)$ factor that should appear when applying the chain rule and power rule correctly.

Step 4: Impact on the Final Derivative

This omission leads to:

• A numerator of $-1 + 36x^8\csc(u)\cot(u)$ instead of the correct $-1 + 36x^8\csc^4(u)\cot(u)$.
• A misalignment in the degrees of the trigonometric functions, affecting the entire expression.

Additionally, in the user's denominator, they have $x + [\csc(u)]^5$ instead of $[x + \csc^4(u)]^2$. This suggests another error:

• The user may have incorrectly squared $x + \csc^4(u)$ to get $x + \csc^5(u)$, possibly by mistakenly adding the exponents or misapplying exponent rules.

Conclusion:

The likely error leading to the incorrect answer is misapplying the chain rule and power rule when differentiating $\csc^4(x^9 - 1)$, resulting in missing exponents on the $\csc(u)$ term and incorrect manipulation of the denominator.

Answer:

The error likely occurred in incorrectly differentiating $\csc^4(x^9 - 1)$ by missing exponents—specifically, treating $\csc^4(u)$ as $\csc(u)$ during differentiation—which led to an incorrect derivative and the wrong final expression.