Gegee:
$f(x) = 2 - 3x^2$
Bepaal $f'(x)$ vanuit eerste beginsels - NSC Mathematics - Question 7 - 2018 - Paper 1
Question 7
Gegee:
$f(x) = 2 - 3x^2$
Bepaal $f'(x)$ vanuit eerste beginsels.
Bepaal:
7.2
7.2.1 $D_{f}[4x + 5]^2$
7.2.2 $\frac{dy}{dx}$ indien $y = \sqrt{x^2 - 8} ... show full transcript
Worked Solution & Example Answer:Gegee:
$f(x) = 2 - 3x^2$
Bepaal $f'(x)$ vanuit eerste beginsels - NSC Mathematics - Question 7 - 2018 - Paper 1
Bepaal $f'(x)$ vanuit eerste beginsels.
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To determine the derivative of the function using first principles, we apply the limit definition of the derivative:
f′(x)=limh→0hf(x+h)−f(x)
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Substitute f(x+h) and f(x):
f(x+h)=2−3(x+h)2=2−3(x2+2xh+h2)=2−3x2−6xh−3h2
Therefore,
f′(x)=limh→0h(2−3x2−6xh−3h2)−(2−3x2)
Which simplifies to:
f′(x)=limh→0h−6xh−3h2
Cancelling h gives:
f′(x)=limh→0(−6x−3h)
Finally, taking the limit as h approaches 0 yields:
f′(x)=−6x
Bepaal $D_{f}[4x + 5]^2$.
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To find the derivative of Df[4x+5]2, we utilize the chain rule:
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First, calculate the derivative of the outer function and the inner function:
If u=4x+5, then Df[u2]=2u⋅dxdu.
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Deriving, we find:
Df[4x+5]2=16x2+40x+25
Thus:
Df[4x+5]2=32x+40.
$\frac{dy}{dx}$ indien $y = \sqrt{x^2 - 8} \frac{x^2}{x^2}$.
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To find rac{dy}{dx} for the function given:
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First, express the function clearly by simplifying:
If y=x2−8⋅x2x2, note that rac{x^2}{x^2}=1 for x=0.
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Thus,
y=x2−8
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Using implicit differentiation:
dxdy=2x2−81⋅2x=x2−8x.
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