Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Certainly! Let's find the third derivative of the function \( f(x) = 4x^4 + 2x^3 - 6x + 1 \) step-by-step.
### Step 1: Find the First Derivative
To find the first derivative of the function \( f(x) \), we apply the power rule of differentiation to each term:
[tex]\[ f(x) = 4x^4 + 2x^3 - 6x + 1 \][/tex]
The derivatives are:
- The derivative of \( 4x^4 \) is \( 16x^3 \)
- The derivative of \( 2x^3 \) is \( 6x^2 \)
- The derivative of \( -6x \) is \( -6 \)
- The derivative of the constant \( 1 \) is \( 0 \)
Combining these results, the first derivative \( f'(x) \) is:
[tex]\[ f'(x) = 16x^3 + 6x^2 - 6 \][/tex]
### Step 2: Find the Second Derivative
Next, we find the second derivative by differentiating \( f'(x) \):
[tex]\[ f'(x) = 16x^3 + 6x^2 - 6 \][/tex]
The derivatives are:
- The derivative of \( 16x^3 \) is \( 48x^2 \)
- The derivative of \( 6x^2 \) is \( 12x \)
- The derivative of the constant \( -6 \) is \( 0 \)
Combining these results, the second derivative \( f''(x) \) is:
[tex]\[ f''(x) = 48x^2 + 12x \][/tex]
### Step 3: Find the Third Derivative
Lastly, we find the third derivative by differentiating \( f''(x) \):
[tex]\[ f''(x) = 48x^2 + 12x \][/tex]
The derivatives are:
- The derivative of \( 48x^2 \) is \( 96x \)
- The derivative of \( 12x \) is \( 12 \)
Combining these results, the third derivative \( f'''(x) \) is:
[tex]\[ f'''(x) = 96x + 12 \][/tex]
### Final Result
Thus, the third derivative of the function \( f(x) = 4x^4 + 2x^3 - 6x + 1 \) is:
[tex]\[ f'''(x) = 96x + 12 \][/tex]
Given [tex]\( y = 12 \)[/tex], the value of the third derivative in terms of [tex]\( x \)[/tex] does not depend on [tex]\( y \)[/tex]. So our final expression remains [tex]\( 96x + 12 \)[/tex].
### Step 1: Find the First Derivative
To find the first derivative of the function \( f(x) \), we apply the power rule of differentiation to each term:
[tex]\[ f(x) = 4x^4 + 2x^3 - 6x + 1 \][/tex]
The derivatives are:
- The derivative of \( 4x^4 \) is \( 16x^3 \)
- The derivative of \( 2x^3 \) is \( 6x^2 \)
- The derivative of \( -6x \) is \( -6 \)
- The derivative of the constant \( 1 \) is \( 0 \)
Combining these results, the first derivative \( f'(x) \) is:
[tex]\[ f'(x) = 16x^3 + 6x^2 - 6 \][/tex]
### Step 2: Find the Second Derivative
Next, we find the second derivative by differentiating \( f'(x) \):
[tex]\[ f'(x) = 16x^3 + 6x^2 - 6 \][/tex]
The derivatives are:
- The derivative of \( 16x^3 \) is \( 48x^2 \)
- The derivative of \( 6x^2 \) is \( 12x \)
- The derivative of the constant \( -6 \) is \( 0 \)
Combining these results, the second derivative \( f''(x) \) is:
[tex]\[ f''(x) = 48x^2 + 12x \][/tex]
### Step 3: Find the Third Derivative
Lastly, we find the third derivative by differentiating \( f''(x) \):
[tex]\[ f''(x) = 48x^2 + 12x \][/tex]
The derivatives are:
- The derivative of \( 48x^2 \) is \( 96x \)
- The derivative of \( 12x \) is \( 12 \)
Combining these results, the third derivative \( f'''(x) \) is:
[tex]\[ f'''(x) = 96x + 12 \][/tex]
### Final Result
Thus, the third derivative of the function \( f(x) = 4x^4 + 2x^3 - 6x + 1 \) is:
[tex]\[ f'''(x) = 96x + 12 \][/tex]
Given [tex]\( y = 12 \)[/tex], the value of the third derivative in terms of [tex]\( x \)[/tex] does not depend on [tex]\( y \)[/tex]. So our final expression remains [tex]\( 96x + 12 \)[/tex].
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.