1.
Question 1. Find the equation of the tangent line to \(y=4x\) at \(x=3\).
Step 1: Differentiate \(y=4x\). Step 2: Slope is 4. Step 3: Point is \((3,12)\). Step 4: Use point-slope form to obtain \(y=4x\). Answer: D
2.
Question 2. Find the instantaneous slope of \(y=\frac{1}{x}\) at \(x=2\).
Step 1: Differentiate: \(y'=-1/x^2\). Step 2: Substitute \(x=2\). Step 3: \(y'=-1/4\). Answer: E
3.
Question 3. The derivative of \(y=2x^2+5\) is \(y'=4x\).
Step 1: Differentiate \(2x^2\to4x\). Step 2: Constant derivative is 0. Step 3: Statement is true. Answer: A
4.
Question 4. Find \(dy/dx\) if \(y=\sqrt5\).
Step 1: \(\sqrt5\) is a constant. Step 2: Derivative of a constant is 0. Answer: C
5.
Question 5. If \(h(x)=\frac{x^2}{f(x)}\), \(f(3)=2\), \(f'(3)=4\), then \(h'(3)=-12\).
Step 1: Apply quotient rule. Step 2: Evaluate at \(x=3\). Step 3: \(h'(3)=24/81\neq-12\). Answer: B
6.
Question 6. Find \(dy/dx\) if \(y=7(x^3-4x+3)\).
Step 1: Differentiate inside. Step 2: Multiply by 7. Step 3: \(y'=21x^2-28\). Answer: D
7.
Question 7. If \(y=\frac{4x}{x-3}\), then find \(dy/dx\).
Step 1: Apply quotient rule. Step 2: Simplify. Step 3: \(y'=-12/(x-3)^2\). Answer: B
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