1.
Question 1. \(\frac{d^2}{dx^2}(x^2\cos x)=2\cos x-4x\sin x-x^2\cos x\). True or False?
Step 1: Differentiate \(x^2\cos x\) using the product rule. Step 2: Differentiate again. Step 3: Simplify to obtain \(2\cos x-4x\sin x-x^2\cos x\). Answer: True
2.
Question 2. What is the equation of the tangent line to \(y=x\cos x\) at \(x=\pi\)?
Step 1: Find \(y'=\cos x-x\sin x\). Step 2: At \(x=\pi\), slope \(=-1\). Step 3: Point is \((\pi,-\pi)\). Step 4: Tangent line is \(y=-x\). Answer: \(y=-x\)
3.
Question 3. If \(y=\sqrt{x^2+4}\), then \(y'=\)?
Step 1: Use chain rule. Step 2: Differentiate outer square root. Step 3: Multiply by derivative of inside. Answer: \(\frac{x}{\sqrt{x^2+4}}\)
4.
Question 4. If \(f(x)=(x^3-4)^7\), then \(f'(x)=\)?
Step 1: Apply the appropriate differentiation rule. Step 2: Simplify. Answer: \(21x^2(x^3-4)^6\)
5.
Question 5. If \(y=\cot^{-1}(2x)\), find \(y'\).
Step 1: Apply the appropriate differentiation rule. Step 2: Simplify. Answer: \(-\frac{2}{1+4x^2}\)
6.
Question 6. If \(y=\ln(\cos x)\), find \(\frac{dy}{dx}\).
Step 1: Apply the appropriate differentiation rule. Step 2: Simplify. Answer: \(-\tan x\)
7.
Question 7. If \(y=\cos^{-1}(\sqrt{x})\), stated derivative is correct.
Step 1: Apply the appropriate differentiation rule. Step 2: Simplify. Answer: False
8.
Question 8. If \(x^2+y^2=4\), then \(\frac{dy}{dx}=-\frac{x}{y}\).
Step 1: Apply the appropriate differentiation rule. Step 2: Simplify. Answer: True
9.
Question 9. Find \(\frac{dy}{dx}\) if \(x^3y^2=10\).
Step 1: Apply the appropriate differentiation rule. Step 2: Simplify. Answer: \(-\frac{3y}{2x}\)
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