Find The Derivative Of The Function.$\[ F(t) = (1 + \sqrt{t})(2t^2 - 5) \\]$\[ F^{\prime}(t) = \\]

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the derivative of a given function, which is a crucial skill for any mathematics student.

The Function

The function we will be working with is:

f(t)=(1+t)(2t2−5){ f(t) = (1 + \sqrt{t})(2t^2 - 5) }

This function is a product of two functions, which makes it a good candidate for the product rule of differentiation.

The Product Rule

The product rule states that if we have a function of the form:

f(x)=u(x)v(x){ f(x) = u(x)v(x) }

then the derivative of f(x) is given by:

f′(x)=u′(x)v(x)+u(x)v′(x){ f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x) }

In our case, we have:

u(t)=1+t{ u(t) = 1 + \sqrt{t} }

and

v(t)=2t2−5{ v(t) = 2t^2 - 5 }

Finding the Derivatives of u(t) and v(t)

To apply the product rule, we need to find the derivatives of u(t) and v(t).

The derivative of u(t) is:

u′(t)=12t{ u^{\prime}(t) = \frac{1}{2\sqrt{t}} }

This can be found using the chain rule and the fact that the derivative of √t is 1/(2√t).

The derivative of v(t) is:

v′(t)=4t{ v^{\prime}(t) = 4t }

This can be found using the power rule of differentiation.

Applying the Product Rule

Now that we have found the derivatives of u(t) and v(t), we can apply the product rule to find the derivative of f(t).

f′(t)=u′(t)v(t)+u(t)v′(t){ f^{\prime}(t) = u^{\prime}(t)v(t) + u(t)v^{\prime}(t) }

f′(t)=12t(2t2−5)+(1+t)(4t){ f^{\prime}(t) = \frac{1}{2\sqrt{t}}(2t^2 - 5) + (1 + \sqrt{t})(4t) }

Simplifying the Expression

To simplify the expression, we can start by multiplying the terms:

f′(t)=2t2−52t+4t+4tt{ f^{\prime}(t) = \frac{2t^2 - 5}{2\sqrt{t}} + 4t + 4t\sqrt{t} }

Next, we can rationalize the denominator by multiplying the numerator and denominator by 2√t:

f′(t)=2t2−52t×2t2t+4t+4tt{ f^{\prime}(t) = \frac{2t^2 - 5}{2\sqrt{t}} \times \frac{2\sqrt{t}}{2\sqrt{t}} + 4t + 4t\sqrt{t} }

f′(t)=2t2−52+4t+4tt{ f^{\prime}(t) = \frac{2t^2 - 5}{2} + 4t + 4t\sqrt{t} }

Final Answer

The final answer is:

f′(t)=t−52+4t+4tt{ f^{\prime}(t) = t - \frac{5}{2} + 4t + 4t\sqrt{t} }

f′(t)=9t2−52+4tt{ f^{\prime}(t) = \frac{9t}{2} - \frac{5}{2} + 4t\sqrt{t} }

Conclusion

In this article, we have found the derivative of a given function using the product rule of differentiation. We have also simplified the expression to obtain the final answer. The product rule is a powerful tool for finding the derivatives of functions that are products of other functions. With practice and experience, you will become proficient in applying the product rule and other rules of differentiation to find the derivatives of various functions.

Common Mistakes to Avoid

When finding the derivative of a function using the product rule, it is easy to make mistakes. Here are some common mistakes to avoid:

  • Not identifying the correct functions: Make sure to identify the correct functions u(t) and v(t) before applying the product rule.
  • Not finding the derivatives of u(t) and v(t): Make sure to find the derivatives of u(t) and v(t) before applying the product rule.
  • Not applying the product rule correctly: Make sure to apply the product rule correctly by multiplying the derivatives of u(t) and v(t) and adding the result to the product of u(t) and v'(t).

Practice Problems

To practice finding the derivatives of functions using the product rule, try the following problems:

  • Find the derivative of the function f(x) = (x + 1)(2x^2 - 3)
  • Find the derivative of the function f(x) = (x - 2)(x^2 + 1)
  • Find the derivative of the function f(x) = (x + √x)(x^2 - 2)

References

  • Calculus by Michael Spivak
  • Calculus by James Stewart
  • Calculus by David Guichard

Glossary

  • Derivative: The rate of change of a function with respect to its input.
  • Product rule: A rule for finding the derivative of a function that is a product of two functions.
  • Chain rule: A rule for finding the derivative of a composite function.
  • Power rule: A rule for finding the derivative of a function that is a power of x.
    Q&A: Finding the Derivative of a Function =============================================

Q: What is the product rule of differentiation?

A: The product rule of differentiation is a rule for finding the derivative of a function that is a product of two functions. It states that if we have a function of the form:

f(x)=u(x)v(x){ f(x) = u(x)v(x) }

then the derivative of f(x) is given by:

f′(x)=u′(x)v(x)+u(x)v′(x){ f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x) }

Q: How do I apply the product rule?

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

  1. Identify the two functions u(x) and v(x) that make up the product.
  2. Find the derivatives of u(x) and v(x).
  3. Multiply the derivative of u(x) by v(x) and add the result to the product of u(x) and the derivative of v(x).

Q: What are some common mistakes to avoid when applying the product rule?

A: Here are some common mistakes to avoid when applying the product rule:

  • Not identifying the correct functions: Make sure to identify the correct functions u(x) and v(x) before applying the product rule.
  • Not finding the derivatives of u(x) and v(x): Make sure to find the derivatives of u(x) and v(x) before applying the product rule.
  • Not applying the product rule correctly: Make sure to apply the product rule correctly by multiplying the derivatives of u(x) and v(x) and adding the result to the product of u(x) and v'(x).

Q: Can I use the product rule to find the derivative of a function that is a product of more than two functions?

A: Yes, you can use the product rule to find the derivative of a function that is a product of more than two functions. However, you will need to apply the product rule multiple times, using the result of each application as one of the functions in the next application.

Q: How do I find the derivative of a function that is a product of a constant and a function?

A: To find the derivative of a function that is a product of a constant and a function, you can use the product rule with the constant as one of the functions. For example, if we have the function:

f(x)=2x2{ f(x) = 2x^2 }

we can find its derivative using the product rule with the constant 2 as one of the functions:

f′(x)=2×2x+x2×0{ f^{\prime}(x) = 2 \times 2x + x^2 \times 0 }

f′(x)=4x{ f^{\prime}(x) = 4x }

Q: Can I use the product rule to find the derivative of a function that is a product of a function and a constant?

A: Yes, you can use the product rule to find the derivative of a function that is a product of a function and a constant. In this case, the constant is treated as one of the functions, and the derivative of the constant is zero.

Q: How do I find the derivative of a function that is a product of two functions that are both functions of x?

A: To find the derivative of a function that is a product of two functions that are both functions of x, you can use the product rule with the two functions as u(x) and v(x). For example, if we have the function:

f(x)=(x+1)(2x2−3){ f(x) = (x + 1)(2x^2 - 3) }

we can find its derivative using the product rule with the two functions as u(x) and v(x):

f′(x)=(x+1)×4x+(2x2−3)×1{ f^{\prime}(x) = (x + 1) \times 4x + (2x^2 - 3) \times 1 }

f′(x)=4x2+4x+2x2−3{ f^{\prime}(x) = 4x^2 + 4x + 2x^2 - 3 }

f′(x)=6x2+4x−3{ f^{\prime}(x) = 6x^2 + 4x - 3 }

Q: Can I use the product rule to find the derivative of a function that is a product of two functions that are both functions of x and y?

A: Yes, you can use the product rule to find the derivative of a function that is a product of two functions that are both functions of x and y. In this case, you will need to use the chain rule in addition to the product rule.

Q: How do I find the derivative of a function that is a product of two functions that are both functions of x and y, using the product rule and the chain rule?

A: To find the derivative of a function that is a product of two functions that are both functions of x and y, using the product rule and the chain rule, you can follow these steps:

  1. Identify the two functions u(x, y) and v(x, y) that make up the product.
  2. Find the partial derivatives of u(x, y) and v(x, y) with respect to x and y.
  3. Apply the product rule to find the derivative of the product of u(x, y) and v(x, y).
  4. Use the chain rule to find the derivative of the result with respect to x and y.

For example, if we have the function:

f(x,y)=(x+y)(2x2−3y){ f(x, y) = (x + y)(2x^2 - 3y) }

we can find its derivative using the product rule and the chain rule:

f′(x,y)=(x+y)×4x+(2x2−3y)×1{ f^{\prime}(x, y) = (x + y) \times 4x + (2x^2 - 3y) \times 1 }

f′(x,y)=4x2+4xy+2x2−3y{ f^{\prime}(x, y) = 4x^2 + 4xy + 2x^2 - 3y }

f′(x,y)=6x2+4xy−3y{ f^{\prime}(x, y) = 6x^2 + 4xy - 3y }

Q: Can I use the product rule to find the derivative of a function that is a product of two functions that are both functions of x, y, and z?

A: Yes, you can use the product rule to find the derivative of a function that is a product of two functions that are both functions of x, y, and z. In this case, you will need to use the chain rule in addition to the product rule.

Q: How do I find the derivative of a function that is a product of two functions that are both functions of x, y, and z, using the product rule and the chain rule?

A: To find the derivative of a function that is a product of two functions that are both functions of x, y, and z, using the product rule and the chain rule, you can follow these steps:

  1. Identify the two functions u(x, y, z) and v(x, y, z) that make up the product.
  2. Find the partial derivatives of u(x, y, z) and v(x, y, z) with respect to x, y, and z.
  3. Apply the product rule to find the derivative of the product of u(x, y, z) and v(x, y, z).
  4. Use the chain rule to find the derivative of the result with respect to x, y, and z.

For example, if we have the function:

f(x,y,z)=(x+y+z)(2x2−3y+4z){ f(x, y, z) = (x + y + z)(2x^2 - 3y + 4z) }

we can find its derivative using the product rule and the chain rule:

f′(x,y,z)=(x+y+z)×4x+(2x2−3y+4z)×1{ f^{\prime}(x, y, z) = (x + y + z) \times 4x + (2x^2 - 3y + 4z) \times 1 }

f′(x,y,z)=4x2+4xy+4xz+2x2−3y+4z{ f^{\prime}(x, y, z) = 4x^2 + 4xy + 4xz + 2x^2 - 3y + 4z }

f′(x,y,z)=6x2+4xy+4xz−3y+4z{ f^{\prime}(x, y, z) = 6x^2 + 4xy + 4xz - 3y + 4z }

Conclusion

In this article, we have answered some common questions about finding the derivative of a function using the product rule. We have also provided examples of how to apply the product rule to find the derivative of a function that is a product of two functions that are both functions of x, y, and z. With practice and experience, you will become proficient in applying the product rule and other rules of differentiation to find the derivatives of various functions.