Differentiating polynomials
Practice Flashcards
State the power rule for differentiation.
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All Flashcards in Topic 5.3
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5.3.19 cards
State the power rule for differentiation.
d/dx(xⁿ) = n·xⁿ⁻¹ — multiply by the power, then reduce the power by 1.
What is the derivative of a constant?
0.
How do you differentiate a·xⁿ (constant multiple)?
a·n·xⁿ⁻¹ — the constant stays and multiplies.
How do you differentiate a polynomial?
Differentiate each term separately (term by term), keeping the signs.
Derivative of 4x?
4 (since 4x = 4x¹ → 4·1·x⁰ = 4).
How do you differentiate 1/xⁿ?
Rewrite as x⁻ⁿ, then apply the power rule.
Derivative of 1/x?
x⁻¹ → −x⁻² = −1/x².
How do you differentiate √x?
Write √x = x^(1/2); derivative ½x^(−1/2) = 1/(2√x).
Common sign slip with negative powers?
Forgetting that subtracting 1 makes the power more negative (e.g. −2 → −3).
5.3.29 cards
How do you find the gradient of a curve at x = a?
Differentiate to get f'(x), then substitute x = a to get f'(a).
Which comes first: differentiate or substitute?
Differentiate first, then substitute the value.
How do you find where a curve has gradient m?
Set f'(x) = m and solve for x.
Why might 'find where the gradient is m' have two answers?
If f'(x) is a quadratic, f'(x) = m can have two solutions.
How is a tangent's gradient related to its angle with the x-axis?
Gradient = tan(angle).
Gradient of a tangent making 45° with the x-axis?
tan 45° = 1.
Where does a curve have a horizontal tangent?
Where f'(x) = 0.
Gradient at a point: substitute into f or f'?
Into f'(x) (the derivative), not f(x).
Find x where y = x² has gradient 8?
2x = 8 ⇒ x = 4.
Topic 5.3 study notes
Full notes & explanations for Differentiating polynomials
Math AA SL exam skills
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