Use The Chain Rule To Find The Derivative Of ${ 8 \sqrt{4x^3 + 10x^4} }$Type Your Answer Without Fractional Or Negative Exponents. Use { \operatorname{sqrt}(x)$}$ For { \sqrt{x}$}$.

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Introduction

The chain rule is a fundamental concept in calculus that allows us to find the derivative of composite functions. In this article, we will use the chain rule to find the derivative of the function ${ 8 \sqrt{4x^3 + 10x^4} }$. This function is a composite function, meaning it is the result of combining two or more functions. The chain rule enables us to find the derivative of such functions by breaking them down into smaller components.

Understanding the Chain Rule

The chain rule states that if we have a composite function of the form f(g(x))f(g(x)), then the derivative of this function is given by f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x). In other words, we first find the derivative of the outer function ff, and then multiply it by the derivative of the inner function gg. This rule is essential in finding the derivatives of complex functions.

Applying the Chain Rule to the Given Function

To find the derivative of the function $ 8 \sqrt{4x^3 + 10x^4} }$, we can break it down into two components the outer function $\sqrt{u$ and the inner function u=4x3+10x4u = 4x^3 + 10x^4. We will first find the derivative of the outer function, and then multiply it by the derivative of the inner function.

Derivative of the Outer Function

The derivative of the outer function u\sqrt{u} is given by 12u\frac{1}{2\sqrt{u}}. This is because the derivative of the square root function is 12x\frac{1}{2\sqrt{x}}.

Derivative of the Inner Function

The derivative of the inner function u=4x3+10x4u = 4x^3 + 10x^4 is given by dudx=12x2+40x3\frac{du}{dx} = 12x^2 + 40x^3. This is because the derivative of a polynomial function is the sum of the derivatives of its terms.

Applying the Chain Rule

Now that we have found the derivatives of the outer and inner functions, we can apply the chain rule to find the derivative of the given function. We multiply the derivative of the outer function by the derivative of the inner function:

ddx(84x3+10x4)=124x3+10x4⋅(12x2+40x3)\frac{d}{dx} \left( 8 \sqrt{4x^3 + 10x^4} \right) = \frac{1}{2\sqrt{4x^3 + 10x^4}} \cdot (12x^2 + 40x^3)

Simplifying the Derivative

We can simplify the derivative by combining the terms:

ddx(84x3+10x4)=12x2+40x324x3+10x4\frac{d}{dx} \left( 8 \sqrt{4x^3 + 10x^4} \right) = \frac{12x^2 + 40x^3}{2\sqrt{4x^3 + 10x^4}}

Conclusion

In this article, we used the chain rule to find the derivative of the function ${ 8 \sqrt{4x^3 + 10x^4} }$. We broke down the function into two components, found the derivatives of the outer and inner functions, and then applied the chain rule to find the derivative of the given function. The chain rule is a powerful tool in calculus that enables us to find the derivatives of complex functions.

Example Problems

Problem 1

Find the derivative of the function ${ 3 \sqrt{2x^2 + 5x} }$.

Solution

To find the derivative of the function $ 3 \sqrt{2x^2 + 5x} }$, we can break it down into two components the outer function $\sqrt{u$ and the inner function u=2x2+5xu = 2x^2 + 5x. We will first find the derivative of the outer function, and then multiply it by the derivative of the inner function.

The derivative of the outer function u\sqrt{u} is given by 12u\frac{1}{2\sqrt{u}}. The derivative of the inner function u=2x2+5xu = 2x^2 + 5x is given by dudx=4x+5\frac{du}{dx} = 4x + 5.

We can apply the chain rule to find the derivative of the given function:

ddx(32x2+5x)=122x2+5x⋅(4x+5)\frac{d}{dx} \left( 3 \sqrt{2x^2 + 5x} \right) = \frac{1}{2\sqrt{2x^2 + 5x}} \cdot (4x + 5)

Problem 2

Find the derivative of the function ${ 2 \sqrt{3x^2 - 2x} }$.

Solution

To find the derivative of the function $ 2 \sqrt{3x^2 - 2x} }$, we can break it down into two components the outer function $\sqrt{u$ and the inner function u=3x2−2xu = 3x^2 - 2x. We will first find the derivative of the outer function, and then multiply it by the derivative of the inner function.

The derivative of the outer function u\sqrt{u} is given by 12u\frac{1}{2\sqrt{u}}. The derivative of the inner function u=3x2−2xu = 3x^2 - 2x is given by dudx=6x−2\frac{du}{dx} = 6x - 2.

We can apply the chain rule to find the derivative of the given function:

ddx(23x2−2x)=123x2−2x⋅(6x−2)\frac{d}{dx} \left( 2 \sqrt{3x^2 - 2x} \right) = \frac{1}{2\sqrt{3x^2 - 2x}} \cdot (6x - 2)

Final Answer

The final answer is 12x2+40x324x3+10x4\boxed{\frac{12x^2 + 40x^3}{2\sqrt{4x^3 + 10x^4}}}.

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Q: What is the chain rule in calculus?

A: The chain rule is a fundamental concept in calculus that allows us to find the derivative of composite functions. It states that if we have a composite function of the form f(g(x))f(g(x)), then the derivative of this function is given by f′(g(x))⋅g′(x)f'(g(x)) \cdot g'(x).

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

A: To apply the chain rule, you need to break down the composite function into two components: the outer function and the inner function. Then, you find the derivatives of the outer and inner functions separately. Finally, you multiply the derivatives of the outer and inner functions to get the derivative of the composite function.

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

A: Some common mistakes to avoid when applying the chain rule include:

  • Forgetting to break down the composite function into two components
  • Not finding the derivatives of the outer and inner functions separately
  • Not multiplying the derivatives of the outer and inner functions to get the derivative of the composite function
  • Not simplifying the derivative to its simplest form

Q: Can I use the chain rule to find the derivative of any composite function?

A: Yes, you can use the chain rule to find the derivative of any composite function. However, you need to make sure that the composite function is in the form f(g(x))f(g(x)), where ff and gg are both functions of xx.

Q: How do I know if a function is a composite function?

A: A function is a composite function if it can be written in the form f(g(x))f(g(x)), where ff and gg are both functions of xx. For example, the function f(x)=2x2+5xf(x) = \sqrt{2x^2 + 5x} is a composite function because it can be written as f(g(x))f(g(x)), where g(x)=2x2+5xg(x) = 2x^2 + 5x and f(x)=xf(x) = \sqrt{x}.

Q: Can I use the chain rule to find the derivative of a function that is not a composite function?

A: No, you cannot use the chain rule to find the derivative of a function that is not a composite function. The chain rule is only applicable to composite functions, which are functions that can be written in the form f(g(x))f(g(x)), where ff and gg are both functions of xx.

Q: How do I simplify the derivative of a composite function?

A: To simplify the derivative of a composite function, you need to combine the terms and cancel out any common factors. You can also use algebraic manipulations to simplify the derivative.

Q: Can I use the chain rule to find the derivative of a function that has multiple composite functions?

A: Yes, you can use the chain rule to find the derivative of a function that has multiple composite functions. However, you need to apply the chain rule multiple times to find the derivative of each composite function.

Q: How do I know if a function has multiple composite functions?

A: A function has multiple composite functions if it can be written in the form f(g(h(x)))f(g(h(x))), where ff, gg, and hh are all functions of xx. For example, the function f(x)=23x2−2xf(x) = \sqrt{2\sqrt{3x^2 - 2x}} has multiple composite functions because it can be written as f(g(h(x)))f(g(h(x))), where h(x)=3x2−2xh(x) = 3x^2 - 2x, g(x)=xg(x) = \sqrt{x}, and f(x)=xf(x) = \sqrt{x}.

Q: Can I use the chain rule to find the derivative of a function that has a trigonometric function?

A: Yes, you can use the chain rule to find the derivative of a function that has a trigonometric function. However, you need to use the chain rule in conjunction with the derivatives of the trigonometric functions.

Q: How do I find the derivative of a trigonometric function?

A: The derivatives of the trigonometric functions are:

  • ddxsin⁡x=cos⁡x\frac{d}{dx} \sin x = \cos x
  • ddxcos⁡x=−sin⁡x\frac{d}{dx} \cos x = -\sin x
  • ddxtan⁡x=sec⁡2x\frac{d}{dx} \tan x = \sec^2 x
  • ddxcsc⁡x=−csc⁡xcot⁡x\frac{d}{dx} \csc x = -\csc x \cot x
  • ddxsec⁡x=sec⁡xtan⁡x\frac{d}{dx} \sec x = \sec x \tan x
  • ddxcot⁡x=−csc⁡2x\frac{d}{dx} \cot x = -\csc^2 x

Q: Can I use the chain rule to find the derivative of a function that has a logarithmic function?

A: Yes, you can use the chain rule to find the derivative of a function that has a logarithmic function. However, you need to use the chain rule in conjunction with the derivatives of the logarithmic functions.

Q: How do I find the derivative of a logarithmic function?

A: The derivatives of the logarithmic functions are:

  • ddxln⁡x=1x\frac{d}{dx} \ln x = \frac{1}{x}
  • ddxlog⁡ax=1xln⁡a\frac{d}{dx} \log_a x = \frac{1}{x \ln a}

Q: Can I use the chain rule to find the derivative of a function that has an exponential function?

A: Yes, you can use the chain rule to find the derivative of a function that has an exponential function. However, you need to use the chain rule in conjunction with the derivatives of the exponential functions.

Q: How do I find the derivative of an exponential function?

A: The derivatives of the exponential functions are:

  • ddxex=ex\frac{d}{dx} e^x = e^x
  • ddxax=axln⁡a\frac{d}{dx} a^x = a^x \ln a

Q: Can I use the chain rule to find the derivative of a function that has a power function?

A: Yes, you can use the chain rule to find the derivative of a function that has a power function. However, you need to use the chain rule in conjunction with the derivatives of the power functions.

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

A: The derivatives of the power functions are:

  • ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}

Q: Can I use the chain rule to find the derivative of a function that has a rational function?

A: Yes, you can use the chain rule to find the derivative of a function that has a rational function. However, you need to use the chain rule in conjunction with the derivatives of the rational functions.

Q: How do I find the derivative of a rational function?

A: The derivatives of the rational functions are:

  • ddxf(x)g(x)=f′(x)g(x)−f(x)g′(x)g(x)2\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}

Q: Can I use the chain rule to find the derivative of a function that has an absolute value function?

A: Yes, you can use the chain rule to find the derivative of a function that has an absolute value function. However, you need to use the chain rule in conjunction with the derivatives of the absolute value functions.

Q: How do I find the derivative of an absolute value function?

A: The derivatives of the absolute value functions are:

  • ddx∣x∣=x∣x∣\frac{d}{dx} |x| = \frac{x}{|x|}