Find The Derivative Of The Function:$\[ R = \frac{2}{s^3} - \frac{7}{s} \\]A) \[$-\frac{6}{s^4} + \frac{7}{s^2}\$\]B) \[$\frac{2}{5^4} - \frac{7}{3^2}\$\]C) \[$-\frac{6}{s^2} + \frac{7}{s^2}\$\]D) \[$\frac{6}{5^4} -

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a given function, which is a crucial concept in mathematics and has numerous applications in various fields. We will use the power rule and the quotient rule to find the derivative of the function.

The Function

The given function is:

r=2s3−7s{ r = \frac{2}{s^3} - \frac{7}{s} }

Step 1: Apply the Quotient Rule

To find the derivative of the function, we will apply the quotient rule, which states that if we have a function of the form:

f(x)=g(x)h(x){ f(x) = \frac{g(x)}{h(x)} }

then the derivative of the function is:

f′(x)=h(x)g′(x)−g(x)h′(x)[h(x)]2{ f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} }

In our case, we have:

r=2s3−7s{ r = \frac{2}{s^3} - \frac{7}{s} }

We can rewrite the function as:

r=2s3−7s0s{ r = \frac{2}{s^3} - \frac{7s^0}{s} }

Now, we can apply the quotient rule to find the derivative of the function.

Step 2: Find the Derivative of the Function

Using the quotient rule, we get:

r′=s3⋅dds(2s3)−2s3⋅dds(s3)(s3)2−s⋅dds(7s0s)−7s0s⋅dds(s)s2{ r' = \frac{s^3 \cdot \frac{d}{ds} \left( \frac{2}{s^3} \right) - \frac{2}{s^3} \cdot \frac{d}{ds} (s^3)}{(s^3)^2} - \frac{s \cdot \frac{d}{ds} \left( \frac{7s^0}{s} \right) - \frac{7s^0}{s} \cdot \frac{d}{ds} (s)}{s^2} }

Simplifying the expression, we get:

r′=s3⋅(−6s4)−2s3⋅3s2(s3)2−s⋅(−7s)−7s0s⋅1s2{ r' = \frac{s^3 \cdot \left( -\frac{6}{s^4} \right) - \frac{2}{s^3} \cdot 3s^2}{(s^3)^2} - \frac{s \cdot \left( -\frac{7}{s} \right) - \frac{7s^0}{s} \cdot 1}{s^2} }

r′=−6s3s4−6s2s3s6+7s−7ss{ r' = \frac{-\frac{6s^3}{s^4} - \frac{6s^2}{s^3}}{s^6} + \frac{\frac{7}{s} - \frac{7}{s}}{s} }

r′=−6s−6ss3{ r' = \frac{-\frac{6}{s} - \frac{6}{s}}{s^3} }

r′=−12ss3{ r' = \frac{-\frac{12}{s}}{s^3} }

r′=−12s4{ r' = -\frac{12}{s^4} }

However, we are not done yet. We still need to find the derivative of the second term in the original function.

Step 3: Find the Derivative of the Second Term

The second term in the original function is:

−7s{ -\frac{7}{s} }

Using the power rule, we get:

dds(−7s)=−7s2{ \frac{d}{ds} \left( -\frac{7}{s} \right) = -\frac{7}{s^2} }

Now, we can substitute this result back into the expression for the derivative of the function.

Step 4: Combine the Results

Substituting the result from Step 3 back into the expression for the derivative of the function, we get:

r′=−12s4+7s2{ r' = -\frac{12}{s^4} + \frac{7}{s^2} }

Therefore, the derivative of the function is:

r′=−12s4+7s2{ r' = -\frac{12}{s^4} + \frac{7}{s^2} }

Conclusion

In this article, we found the derivative of a given function using the quotient rule and the power rule. We applied the quotient rule to find the derivative of the first term in the function and then used the power rule to find the derivative of the second term. Finally, we combined the results to get the final derivative of the function.

Answer

The correct answer is:

Introduction

In our previous article, we found the derivative of a given function using the quotient rule and the power rule. In this article, we will provide a Q&A guide to help you understand the concept of derivatives and how to apply the rules to find the derivative of a function.

Q: What is the derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to one of its variables. It is a measure of how fast the function changes as the variable changes.

Q: What are the rules for finding the derivative of a function?

A: There are two main rules for finding the derivative of a function:

  1. Power Rule: If we have a function of the form:

f(x)=xn{ f(x) = x^n }

then the derivative of the function is:

f′(x)=nxn−1{ f'(x) = nx^{n-1} }

  1. Quotient Rule: If we have a function of the form:

f(x)=g(x)h(x){ f(x) = \frac{g(x)}{h(x)} }

then the derivative of the function is:

f′(x)=h(x)g′(x)−g(x)h′(x)[h(x)]2{ f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{[h(x)]^2} }

Q: How do I apply the power rule to find the derivative of a function?

A: To apply the power rule, you need to follow these steps:

  1. Identify the variable and the exponent in the function.
  2. Multiply the exponent by the coefficient of the variable.
  3. Subtract 1 from the exponent.
  4. Write the result as the derivative of the function.

Q: How do I apply the quotient rule to find the derivative of a function?

A: To apply the quotient rule, you need to follow these steps:

  1. Identify the numerator and the denominator in the function.
  2. Find the derivative of the numerator and the denominator separately.
  3. Substitute the derivatives back into the quotient rule formula.
  4. Simplify the expression to get the final derivative of the function.

Q: What are some common mistakes to avoid when finding the derivative of a function?

A: Here are some common mistakes to avoid when finding the derivative of a function:

  1. Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying the expression.
  2. Not using the correct rules: Make sure to use the correct rules (power rule and quotient rule) to find the derivative of the function.
  3. Not checking the units: Make sure to check the units of the function and the derivative to ensure that they are consistent.

Q: How do I check my work when finding the derivative of a function?

A: Here are some steps to check your work when finding the derivative of a function:

  1. Re-read the problem: Make sure you understand the problem and the function.
  2. Check the units: Make sure the units of the function and the derivative are consistent.
  3. Simplify the expression: Simplify the expression to ensure that it is correct.
  4. Check the final answer: Check the final answer to ensure that it is correct.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of derivatives and how to apply the rules to find the derivative of a function. We covered the power rule and the quotient rule, and provided tips on how to avoid common mistakes and check your work. We hope this guide has been helpful in your understanding of derivatives.