Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given
y=3cot^(3)(x), find
(dy)/(dx).
Answer:
(dy)/(dx)=

Given $y=3 \cot ^{3}(x)$ , find $\frac{d y}{d x}$ . $\newline$ Answer: $\frac{d y}{d x}=$

View step-by-step help

View step-by-step help

Write a formula for an arithmetic sequence

Full solution

Q. Given

y=3 \cot ^{3}(x)

, find

\frac{d y}{d x}

.

\newline

Answer:

\frac{d y}{d x}=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $y = 3\cot^{3}(x)$ , which means $y$ is equal to three times the cube of the cotangent of $x$ .
Apply Chain Rule: Apply the chain rule for differentiation. $\newline$ The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $f(u) = 3u^3$ and the inner function is $u(x) = \cot(x)$ . We need to find the derivative of both functions.
Differentiate Outer Function: Differentiate the outer function $f(u) = 3u^3$ with respect to $u$ . The derivative of $f(u)$ with respect to $u$ is $f'(u) = 3 \times 3u^{3-1} = 9u^2$ .
Differentiate Inner Function: Differentiate the inner function $u(x) = \cot(x)$ with respect to $x$ . The derivative of $\cot(x)$ with respect to $x$ is $-\csc^2(x)$ , where $\csc(x)$ is the cosecant of $x$ .
Apply Chain Rule with Derivatives: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ $\frac{dy}{dx} = f'(u) \cdot \frac{du}{dx} = 9u^2 \cdot (-\csc^2(x)) = 9\cot^2(x) \cdot (-\csc^2(x))$ .
Substitute Back to x: Substitute $u$ back with $\cot(x)$ to get the derivative in terms of $x$ . $\frac{dy}{dx} = 9\cot^2(x) \cdot (-\csc^2(x)) = -9\cot^2(x)\csc^2(x)$ .
Check for Errors: Check for any mathematical errors in the differentiation process. $\newline$ No mathematical errors were made in the differentiation process.

More problems from Write a formula for an arithmetic sequence

Question

Find the sum of the finite arithmetic series.

\sum_{n=1}^{10} (7n+4)

\newline

Posted 3 months ago

Question

Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`

Posted 2 months ago

Question

Type the missing number in this sequence: `2, 4 8`, ______,`32`

Posted 2 months ago

Question

What kind of sequence is this?

2, 10, 50, 250, \ldots

Choices:Choices:

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

What is the missing number in this pattern?

1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_

Posted 2 months ago

Question

Classify the series.

\sum_{n = 0}^{12} (n + 2)^3

\newline

\newline

\text{[A]arithmetic}

\newline

\text{[B]geometric}

\newline

\text{[C]both}

\newline

\text{[D]neither}

Posted 2 months ago

Question

Find the first three partial sums of the series.

\newline

1 + 6 + 11 + 16 + 21 + 26 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 2 months ago

Question

Find the third partial sum of the series.

\newline

3 + 9 + 15 + 21 + 27 + 33 + \cdots

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

S_3 =

Posted 2 months ago

Question

Find the first three partial sums of the series.

\newline

1 + 7 + 13 + 19 + 25 + 31 + \cdots

\newline

Write your answers as integers or fractions in simplest form.

\newline

S_1 =

\newline

S_2 =

\newline

S_3 =

Posted 3 months ago

Question

Does the infinite geometric series converge or diverge?

\newline

1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots

\newline

\newline

\text{[A]converge}

\newline

\text{[B]diverge}

Posted 2 months ago