1.
Question 1. The derivative of \(y=\sqrt{x}\) is
Step 1: Rewrite the square root as a power: \(y=\sqrt{x}=x^{1/2}\). Step 2: Use the power rule: \(\frac{d}{dx}x^n=nx^{n-1}\). Step 3: Differentiate: \(y'=\frac{1}{2}x^{-1/2}\). Step 4: Rewrite using a square root: \(y'=\frac{1}{2\sqrt{x}}\). Answer: B. \(y'=\frac{1}{2\sqrt{x}}\)
2.
Question 2. \(\lim_{h\to0}\frac{\sqrt{2+h}-\sqrt{2}}{h}\) represents the derivative of \(f(x)=\sqrt{x}\) at \(x=a\). What is \(a\)?
Step 1: Recall the definition of the derivative at \(x=a\): \(f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}\). Step 2: For \(f(x)=\sqrt{x}\), this becomes \(f'(a)=\lim_{h\to0}\frac{\sqrt{a+h}-\sqrt{a}}{h}\). Step 3: Compare this with \(\lim_{h\to0}\frac{\sqrt{2+h}-\sqrt{2}}{h}\). Step 4: We see that \(a=2\). Answer: A. \(2\)
3.
Question 3. The derivative of \(y=2x^2+5\) is \(y'=4x\).
Step 1: Differentiate \(y=2x^2+5\). Step 2: \(\frac{d}{dx}(2x^2)=4x\). Step 3: \(\frac{d}{dx}(5)=0\). Step 4: Therefore, \(y'=4x+0=4x\). Step 5: The statement is true. Answer: B. True
4.
Question 4. If \(y=\frac{8}{\sqrt{x}}\), find \(y'(4)\).
Step 1: Rewrite the function as \(y=8x^{-1/2}\). Step 2: Differentiate using the power rule: \(y'=8\left(-\frac{1}{2}\right)x^{-3/2}\). Step 3: Simplify: \(y'=-4x^{-3/2}=\frac{-4}{x^{3/2}}\). Step 4: Substitute \(x=4\): \(y'(4)=\frac{-4}{4^{3/2}}\). Step 5: Since \(4^{3/2}=8\), \(y'(4)=-\frac{4}{8}=-\frac{1}{2}\). Answer: C. \(-\frac{1}{2}\)
5.
Question 5. Find \(\frac{dy}{dx}\) if \(y=7(x^3-4x+3)\).
Step 1: Use the constant multiple rule: \(y=7(x^3-4x+3)\). Step 2: Differentiate inside the parentheses: \(\frac{d}{dx}(x^3-4x+3)=3x^2-4\). Step 3: Multiply by \(7\): \(y'=7(3x^2-4)\). Step 4: Simplify: \(y'=21x^2-28\). Answer: D. \(21x^2-28\)
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Question 6. If \(f(2)=3\), \(f'(2)=-2\), \(g(2)=4\), and \(g'(2)=-3\), find \(h'(2)\) if \(h(x)=f(x)\times g(x)\).
Step 1: Use the product rule: \(h'(x)=f'(x)g(x)+f(x)g'(x)\). Step 2: Substitute \(x=2\): \(h'(2)=f'(2)g(2)+f(2)g'(2)\). Step 3: Substitute the given values: \(h'(2)=(-2)(4)+(3)(-3)\). Step 4: Simplify: \(h'(2)=-8-9=-17\). Answer: A. \(-17\)
7.
Question 7. If \(y=5x^3+10x^2-15\), find \(\frac{d^2y}{dx^2}\).
Step 1: Find the first derivative of \(y=5x^3+10x^2-15\). Step 2: \(y'=15x^2+20x\). Step 3: Differentiate again to find the second derivative. Step 4: \(\frac{d^2y}{dx^2}=30x+20\). Answer: B. \(30x+20\)
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