📝 Unit 2: Derivatives
Practice Quiz — Power, Product, Quotient, Chain, Implicit
🔄 Not Graded — Unlimited Retakes
Purpose: Self-check fluency with all major derivative rules. Retake until confident.
Score: 0 / 12
Question 1
Differentiate \(f(x)=4x^5-3x^2+7\). Find \(f'(1)\).
Solution:
\(f'(x)=20x^4-6x\); \(f'(1)=20-6=14\).
Question 2
Find \(\frac{d}{dx}\left(\sqrt{x}+\frac{1}{x^2}\right)\) at \(x=4\).
Solution:
\(\frac{d}{dx}(x^{1/2}+x^{-2})=\frac{1}{2\sqrt{x}}-\frac{2}{x^3}\). At 4: \(\frac{1}{4}-\frac{2}{64}=0.25-0.03125=0.21875\).
Question 3
If \(y=(2x+1)(x^2-3)\), find \(y'\).
Solution:
\(y'=2(x^2-3)+(2x+1)(2x)=2x^2-6+4x^2+2x=6x^2+2x-6\).
Question 4
For \(f(x)=x^2(x-1)^3\), find \(f'(2)\).
Solution:
\(f'=2x(x-1)^3+x^2\cdot 3(x-1)^2=(x-1)^2[2x(x-1)+3x^2]=(x-1)^2(5x^2-2x)\). At 2: \(1\cdot(20-4)=16\).
Question 5
If \(g(x)=\dfrac{x^2+1}{x-2}\), find \(g'(3)\).
Solution:
\(g'=\frac{2x(x-2)-(x^2+1)(1)}{(x-2)^2}=\frac{x^2-4x-1}{(x-2)^2}\). At 3: \((9-12-1)/1=-4\).
Question 6
Find \(\frac{d}{dx}\frac{3x}{x^2+1}\) at \(x=1\).
Solution:
\(\frac{3(x^2+1)-3x(2x)}{(x^2+1)^2}=\frac{3-3x^2}{(x^2+1)^2}\). At 1: \((3-3)/4=0\).
Question 7
If \(y=(3x^2+1)^4\), find \(\frac{dy}{dx}\) at \(x=1\).
Solution:
\(y'=4(3x^2+1)^3\cdot 6x=24x(3x^2+1)^3\). At 1: \(24\cdot 64=1536\).
Question 8
\(\frac{d}{dx}\sqrt{x^2+5}\) at \(x=2\) equals:
Solution:
\(\frac{2x}{2\sqrt{x^2+5}}=\frac{x}{\sqrt{x^2+5}}\). At 2: \(2/3\).
Question 9
If \(h(x)=(x^2-1)^3(x+2)^2\), the factored form of \(h'(x)\) is \((x^2-1)^2(x+2)\,Q(x)\). Find \(Q(0)\).
Solution:
\(h'=3(x^2-1)^2(2x)(x+2)^2+(x^2-1)^3\cdot 2(x+2)=(x^2-1)^2(x+2)[6x(x+2)+2(x^2-1)]\). \(Q(x)=8x^2+12x-2\); \(Q(0)=-2\).
Question 10
Given \(x^2+y^2=25\), find \(dy/dx\) at \((3,4)\).
Solution:
\(2x+2y\,y'=0\) → \(y'=-x/y=-3/4=-0.75\).
Question 11
Implicitly: \(x^2y+xy^2=6\). Find \(dy/dx\) at \((1,2)\).
Solution:
\(2xy+x^2 y'+y^2+2xy y'=0\). At (1,2): \(4+y'+4+4y'=0\) → \(5y'=-8\) → \(y'=-1.6\).
Question 12
For \(y=u^3\) where \(u=2x+1\), Leibniz/chain gives \(dy/dx\) at \(x=1\). Value?
Solution:
\(\frac{dy}{dx}=3u^2\cdot 2=6(2x+1)^2\). At 1: \(6\cdot 9=54\).
📊 Self-Reflection
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