f'(x) = 3*3x² - 5*2x + 2*1 - 0 - Decision Point
Understanding the Derivative f'(x) = 3(3x²) - 5(2x) + 2(1) - 0: A Step-by-Step Guide
Understanding the Derivative f'(x) = 3(3x²) - 5(2x) + 2(1) - 0: A Step-by-Step Guide
When diving into calculus, one of the most essential concepts is derivatives—mathematical tools used to analyze how functions change. Today, we break down a specific derivative expression:
f’(x) = 3(3x²) − 5(2x) + 2(1) − 0
(Note: This simplifies to a polynomial function derived via differentiation rules.)
Understanding the Context
What Is f'(x)? Breaking Down the Expression
The expression:
f’(x) = 3(3x²) − 5(2x) + 2(1) − 0
is a direct application of differentiation. Let's simplify it step by step.
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Key Insights
Start by expanding each term:
- 3(3x²) = 9x²
- −5(2x) = −10x
- +2(1) = +2
- − 0 = 0
Putting it together:
f’(x) = 9x² − 10x + 2
This derivative represents the slope of the original function’s graph at any point x. It quantifies how fast f(x) is changing, vital in optimization, motion analysis, and real-world modeling.
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How Is f’(x) Derived? The Differentiation Process
Although the simplified form is 9x² − 10x + 2, understanding the derivation process using the power rule reinforces mathematical intuition.
For a general quadratic function:
f(x) = ax² + bx + c
its derivative follows:
f’(x) = 2ax − b
In our specific case, after simplifying:
- a = 9 → 2(9)x = 18x? Wait—hold on!
Wait—let’s clarify carefully.
Original expression:
f’(x) = 3(3x²) − 5(2x) + 2(1) − 0 = 9x² − 10x + 2
If we interpret the original phrasing as applying differentiation to 9x² − 10x + 2, then:
Using derivative rules:
- d/dx [xⁿ] = n xⁿ⁻¹
- d/dx [constant] = 0
So:
- d/dx [9x²] = 18x
- d/dx [−10x] = −10
- d/dx [2] = 0
- d/dx [−0] = 0
Thus:
f’(x) = 18x − 10
