Step-By-Step-Math: Find f'(x) for f(x) = 3x⁴ - 2x² + 7x - 5 | Power Rule for Derivatives
Find f'(x) for f(x) = 3x⁴ - 2x² + 7x - 5 | Power Rule for Derivatives Step by Step
Learning Target: I can find the derivative of a polynomial using the power rule
What You'll Learn
- State the power rule
- Apply the power rule term by term
- Simplify the resulting derivative
Key Terms
derivative, power rule, polynomial, exponent, coefficient
Standards Alignment
- CCSS.MATH.CONTENT.HSF.BF.B.4: Find inverse functions and understand rates of change
Math Practices
- MP2: Reason abstractly and quantitatively
- MP6: Attend to precision
- MP7: Look for and make use of structure
Chapters
- 0:00 How do you find the derivative of a polynomial?
- 0:12 What is the power rule for derivatives?
- 0:24 How do you differentiate 3x⁴?
- 0:36 How do you differentiate -2x²?
- 0:48 How do you differentiate 7x?
- 1:00 What happens to constants when differentiating?
- 1:12 How do you combine all the derivative terms?
- 2:07 Final
- 2:16 Verification
- 2:36 Signoff
▶ Click to Read Full Transcript
[00:00] Let's find the derivative of a polynomial step by step. We are given the function f of x equals 3 x to the fourth minus 2 x squared plus 7 x minus 5. Our goal is to find f prime of x using the power rule. Always feel free to pause the video and try it on your own.
[00:19] The power rule is our main tool. It says: to find the derivative of x to the n, bring the exponent n down as a coefficient, multiply by any existing coefficient, then subtract one from the exponent. Let me show you the formula.
[00:35] Let's work term by term. First term: 3 x to the fourth. Bring the exponent 4 down as a coefficient. Multiply it by the existing coefficient 3. That gives us 12. Then subtract one from the exponent: 4 minus 1 equals 3. So we get 12 x cubed.
[00:57] Second term: negative 2 x squared. Bring the exponent 2 down. Multiply by the coefficient negative 2. That gives us negative 4. Subtract one from the exponent: 2 minus 1 equals 1. So we get negative 4 x to the first power, which is just negative 4 x.
[01:19] Third term: 7 x. Remember that x means x to the first power. Bring the exponent 1 down. Multiply by the coefficient 7. That gives us 7. Subtract one from the exponent: 1 minus 1 equals 0. Any variable to the zero power equals 1, so we get just 7.
[01:41] Finally, the constant term negative 5. The derivative of any constant is always zero because constants don't change. So negative 5 becomes zero.
[01:52] Now let's combine all our results. We found 12 x cubed from the first term, negative 4 x from the second term, 7 from the third term, and 0 from the constant. Adding them together gives us our derivative.
[02:07] Simplifying, we get f prime of x equals 12 x cubed minus 4 x plus 7. This is our final answer.
[02:16] Let's quickly verify by checking each term: The derivative of 3 x to the fourth is indeed 12 x cubed. The derivative of negative 2 x squared is negative 4 x. The derivative of 7 x is 7. And the derivative of negative 5 is 0. Our answer is correct.
[02:36] I hope you found this helpful along your journey to discovering the joy in mathematics. This has been Mister Marx AI. Until next time, keep up the good work!
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