How do you differentiate a polynomial?
संक्षिप्त उत्तर
Differentiate each term separately using the power rule: for axⁿ, multiply by the power and reduce the power by one, giving naxⁿ⁻¹. A term in x becomes a constant and a constant term disappears. So y = 3x⁴ − 2x² + 5x − 7 gives dy/dx = 12x³ − 4x + 5.
स्टेप-बाय-स्टेप
- Make sure every term is written as a power of x. Rewrite roots and fractions first: √x = x^½ and 1/x² = x⁻².
- For each term axⁿ, bring the power down to multiply: n × a.
- Subtract one from the power: xⁿ becomes xⁿ⁻¹.
- A term like 5x becomes 5, and a constant like −7 becomes 0.
- Write the derivative as dy/dx or f′(x).
हल किया गया उदाहरण
Differentiate f(x) = 2x³ − 6x² + 4
- 2x³ gives 3 × 2x² = 6x²
- −6x² gives 2 × (−6)x = −12x
- 4 gives 0
- f′(x) = 6x² − 12x
आम गलतियाँ
- Differentiating before expanding brackets or splitting a fraction. Simplify first.
- Leaving the constant in the answer instead of removing it.
- Getting negative powers wrong: x⁻² becomes −2x⁻³, not −2x⁻¹.
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