Determine $f^{\prime}(x)$ from first principles if $f(x)=-2x^{2}-1$ - NSC Mathematics - Question 7 - 2023 - Paper 1
Question 7
Determine $f^{\prime}(x)$ from first principles if $f(x)=-2x^{2}-1$.
Determine:
7.2.1 $f^{\prime}(x)$, if it is given that $f(x)=-2x^{2}+3x^{2}$
7.2.2 $\frac{dy}... show full transcript
Worked Solution & Example Answer:Determine $f^{\prime}(x)$ from first principles if $f(x)=-2x^{2}-1$ - NSC Mathematics - Question 7 - 2023 - Paper 1
Step 1
Determine $f^{\prime}(x)$ from first principles if $f(x)=-2x^{2}-1$
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Answer
To find the derivative from first principles, we use the definition:
f′(x)=limh→0hf(x+h)−f(x)
Substituting for f(x): f(x+h)=−2(x+h)2−1=−2(x2+2xh+h2)−1=−2x2−4xh−2h2−1
Now, substituting into the derivative formula: f′(x)=limh→0h(−2x2−4xh−2h2−1)−(−2x2−1)
Simplifying: f′(x)=limh→0h−4xh−2h2=limh→0(−4x−2h)
Taking the limit as h→0: f′(x)=−4x
Step 2
Determine $f^{\prime}(x)$, if it is given that $f(x)=-2x^{2}+3x^{2}$
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Answer
We first simplify f(x): f(x)=−2x2+3x2=x2
Now, calculating the derivative: f′(x)=2x
Step 3
$\frac{dy}{dx}$ if $y=2x+\frac{1}{\sqrt{4x}}$
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Answer
To find dxdy, we differentiate each term: y=2x+4x1=2x+2x1
The derivative is: dxdy=2−41x−23
Step 4
Determine the values of $x$ for which $f$ is concave down.
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Answer
A function is concave down when its second derivative is negative.
From the previous step, we have: f′(x)=−4x
Calculating the second derivative: f′′(x)=−4
Since f′′(x) is always negative, f is concave down for all x.
