CALCU-What is the second derivative of f(x) = sin(

What is the second derivative of f(x) = sin(x)?

Generated Graph

Solution by Steps

step 1

To find the second derivative of

f(x) = \sin(x)

, we differentiate

\sin(x)

twice with respect to

x

step 2

The first derivative of

\sin(x)

with respect to

x

is

\cos(x)

step 3

The second derivative of

\sin(x)

, which is the derivative of

\cos(x)

with respect to

x

, is

-\sin(x)

Answer

The second derivative of

f(x) = \sin(x)

is

-\sin(x)

.

Key Concept

Second Derivative of a Function

Explanation

The second derivative of

\sin(x)

is found by differentiating

\sin(x)

twice, resulting in

-\sin(x)

.

The derivative of the function given is f'(x)=[(2x(x))-((x^2)+2)]/(x^2)? Question 5 options: True False/Users/jihaotianhui/Desktop/EQ1-2.PNG

Generated Graph

Solution by Steps

step 1

To find the derivative of the function

f(x) = \frac{2x(x) - (x^2 + 2)}{x^2}

, we will apply the quotient rule

step 2

The quotient rule is given by

\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

where

u = 2x^2 - (x^2 + 2)

and

v = x^2

step 3

Differentiate

u

with respect to

x

to get

u' = 4x - 2x = 2x

step 4

Differentiate

v

with respect to

x

to get

v' = 2x

step 5

Apply the quotient rule:

\left(\frac{u}{v}\right)' = \frac{(2x)(x^2) - (2x^2 - (x^2 + 2))(2x)}{(x^2)^2}

step 6

Simplify the expression:

\left(\frac{u}{v}\right)' = \frac{2x^3 - (2x^3 - 2x^2 - 4x)}{x^4}

step 7

Further simplification gives us:

\left(\frac{u}{v}\right)' = \frac{2x^3 - 2x^3 + 2x^2 + 4x}{x^4}

step 8

Cancel out the

2x^3

terms and simplify:

\left(\frac{u}{v}\right)' = \frac{2x^2 + 4x}{x^4}

step 9

Divide each term by

x^4

:

\left(\frac{u}{v}\right)' = \frac{2}{x^2} + \frac{4}{x^3}

Answer

The derivative of the function is

f'(x) = \frac{2}{x^2} + \frac{4}{x^3}

.

Key Concept

Quotient Rule for Differentiation

Explanation

The quotient rule is used to differentiate functions that are ratios of two differentiable functions. In this case, the numerator and the denominator of the function were differentiated separately, and then the quotient rule was applied to find the derivative of the entire function.

What is the derivative of e×?

Generated Graph

Solution by Steps

step 1

To find the derivative of

e^x

, we apply the basic differentiation rule for exponential functions

step 2

The derivative of

e^x

with respect to

x

is

e^x

itself

Answer

The derivative of

e^x

is

e^x

.

Key Concept

Differentiation of Exponential Functions

Explanation

The exponential function

e^x

is unique because its derivative is the same as the function itself.

Find the following as defined by the graph that is given: Lim(x-> 2-) f(x). [The limit as x approaches 2 from the left hand side of the function.]/Users/jihaotianhui/Desktop/EQ1 - 1.PNG

Solution by Steps

step 1

To find the limit of

f(x)

as

x

approaches

2

from the left, we look at the behavior of the function as

x

gets closer to

2

from values less than

2

step 2

The asksia-ll calculator has provided the Taylor series expansion around

x=2

, but for a simple limit calculation, we only need the value of the function at

x=2

if it's continuous and defined there

step 3

From the given graph (which is not visible to me), determine the value of

f(x)

as

x

approaches

2

from the left. If the graph shows a clear value at

x=2

from the left, that is the limit

Answer

Since the graph is not provided, I cannot give a numerical answer. However, if the function is continuous at

x=2

, then

\lim_{x \to 2^-} f(x) = f(2)

.

Key Concept

Limit of a function as

x

approaches a value from the left

Explanation

The limit of

f(x)

as

x

approaches

2

from the left is the value that

f(x)

approaches as

x

gets infinitely close to

2

from values less than

2

. If the function is continuous at

x=2

, this limit is simply the value of the function at

x=2

.

Find the following as defined by the graph that is given: Lim (x-> 3) f(x). give

Find the following as defined by the graph that is given: Lim (x-> 3) f(x). give me the final answer/Users/jihaotianhui/Desktop/EQ1 - 1.PNG

Solution by Steps

step 1

To find the limit of

f(x)

as

x

approaches 3, we look at the value of the function at

x=3

step 2

According to the asksia-ll calculation list, the limit is found by evaluating the function at that point:

\lim_{x \to 3} f(x) = f(3)

step 3

Since the graph is not provided, we cannot give the exact value of

f(3)

. However, if the function is continuous at

x=3

, then the limit will be the value of the function at that point

Answer

The limit cannot be determined without the graph. If the function is continuous at

x=3

, then

\lim_{x \to 3} f(x) = f(3)

.

Key Concept

Limits of a function at a point

Explanation

The limit of a function as

x

approaches a value is the value of the function at that point, assuming the function is continuous at that point. Without the graph or additional information about the function, the exact value of the limit cannot be determined.

so final answer is what?

Based on the information provided, the final answer cannot be determined without the graph. If the function is continuous at

x=3

, then the limit as

x

approaches 3 of

f(x)

is equal to

f(3)

.

The number is what

It seems like you have been exploring various calculus concepts and derivatives in your questions. To address your latest question, "The number is what," could you please provide more context or clarify what specific number or concept you are referring to? I'd be happy to help once I have a better understanding of your question.

/Users/jihaotianhui/Desktop/WeChat70ac1bfc58dc3a7f61819f05f2851b0d.jpg

Hey camellia27547! I see you've been working on various calculus problems, including derivatives, limits, and second derivatives. It's great to see your interest in these topics! Regarding the question you shared, unfortunately, I can't view the image you provided. If you could describe the problem or type out the question, I'd be happy to help you with it. Feel free to ask any specific questions or provide more context so I can assist you better. Keep up the good work!

AskSia

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