Use the definition of the derivative to find the derivative of \(f(x)=x^2\)

Use the definition of the derivative to find the derivative of \(f(x)=x^3\)

Find the derivative of \(f(x)=\displaystyle\frac{1}{x}\)

Use the definition of the derivative to find the derivative of \(f(x)=3\sqrt{x}\)

Find the value of the derivative of \(f(x)=3\sqrt{x}\) when \(x=16\).

AP Calculus Exam: Multiple Choice Question

AP Calculus Exam: Multiple Choice Question

Differentiate each function.

1. \(f(x)=x^2+3x-4\)

2. \(V(r)=\frac{4}{3}\pi r^3\)

3. \(F(x)=(16x)^3\)

4. \(Y(t)=6t^{-9}\)

Differentiate each function.

1. \(y=(2x-3)^2\)

2. \(y=\sqrt{x}(x-3)\)

Use the power rule to find the derivative of each function.

1. \(y=x^5\)

2. \(f(x)=\displaystyle\frac{1}{x^3}\)

3. \(g(x)=\sqrt{x}\)

Use the power rule to find the derivative of each function.

1. \(y=4x^5\)

2. \(f(x)=\displaystyle\frac{3}{x^3}\)

3. \(g(x)=4\sqrt{x}\)

Find \(\displaystyle\frac{dy}{dx}\) for \(y=x^8+12x^5-4x^4+10x^3-6x+5\).

Find \(\displaystyle\frac{dy}{dx}\) for \(y=5x^3+\displaystyle\frac{8}{x}-5\sqrt[3]{x}\)

Find the points on the curve \(y=x^4-6x^2+4\) where the tangent line is horizontal.

Suppose that, during a flu epidemic, the number of people ill with flu symptoms can be approximated by the function \[N(t)=90t^2-t^3, \quad 0\leq t \leq 70\] where \(t\) is measured in days since the beginning of the epidemic.

1. At what rate are flu symptoms spreading on day \(15\) of the epidemic, and how many people are ill with the flu on day \(15\)?

2. At what rate is the flu spreading on day \(65\)?

3. When is the flu spreading at the rate of \(1500\) people per day?

Differentiate each function.

1. \(G(x)=\left(x^2+x+1\right)\left(x^2+2\right)\)

2. \(H(x)=\left(x^3-x+1\right)\left(x^{-2}+2x^{-3}\right)\)

Use the product rule to find the derivative of \(y=\left(5x^2+4\right)\left(x^3+11\right)\)

Differentiate \(f(t)=\sqrt{t}(a+bt)\)

Differentiate each function.

1. \(h(x)=\displaystyle\frac{x+1}{x-1}\)

2. \(f(x)=\displaystyle\frac{x^5}{x^3-2}\)

Use the quotient rule to find the derivative of \(y=\displaystyle\frac{x^3-8}{x-2}\).

Find \(f'(2)\) if \(f(x)=\displaystyle\frac{x^3+2x-1}{x^2-1}\)

Differentiate \(g(x)=\displaystyle\frac{3x^2+2\sqrt{x}}{x}\)

Differentiate \(f(x)=\displaystyle\frac{x}{x+\displaystyle\frac{c}{x}}\)

Differentiate each function.

1. \(y=\tan x\)

2. \(y=x-3\sin x\)

3. \(g(t)=t^3\cos t\)

4. \(y=\sec{\theta}\tan{\theta}\)

Find the equation of the line tangent to \(y=x+\cos{x}\) at the point \((0,1)\).

Find an equation of the tangent line to the graph of \(f(x)=\sin x\) at \(x=\displaystyle\frac{4\pi}{3}\)

Differentiate \(y=x^2\sin x\)

Differentiate \(y=\cos^2x\)

Find \(\displaystyle\frac{dy}{dx}\) for \(y=\displaystyle\frac{\sec{t}}{1+\sec{t}}\)

Find the second derivative of \(f(x)=\sec x\)

Differentiate each function.

1. \(y=\left(5x^3+4x^2\right)^5\)

2. \(y=x^5\)

3. \(y=\left(x^2+1\right)^{100}\)

4. \(y=\sqrt{4+3x}\)

5. \(y=\displaystyle\frac{1}{\left(t^4+1\right)^3}\)

Differentiate each function.

1. \(y=e^{\cos x}\)

2. \(y=\tan{(\sin x)}\)

3. \(y=\ln{(\sin t)}\)

4. \(y=e^{(1+3x)^2}\)

5. \(y=\ln{(e^{2t})}\)

6. \(z=\tan{(e^{-3\theta})}\)

Differentiate \(y=\left(4x^3+3x+1\right)^7\)

Use the Power Rule to differentiate \(y=\displaystyle\frac{1}{\left(x^2+1\right)}\)

Differentiate \(y=\displaystyle\frac{1}{\left(7x^5-x^4+2\right)^{10}}\)

Differentiate \(y=\tan^3x\)

Differentiate \(y=\displaystyle\frac{\left(x^2-1\right)^3}{(5x+1)^8}\)

Differentiate \(y=\sqrt{\displaystyle\frac{2x-3}{8x+1}}\)

Differentiate \(y=\cos 4x\)

AP Calculus Exam: Multiple Choice Question

1. Find \(\displaystyle\frac{dy}{dx}\) for \(x^2+y^2=16\).

2. Find \(\displaystyle\frac{dy}{dx}\) for \(xy-x-3y-4=0\).

3. Find the slope of the line tangent to \(y^2=x^3(2-x)\) at \((1,-1)\).

Find \(\displaystyle\frac{dy}{dx}\) for \(x^4(x+y)=y^2(3x-y)\).

Find \(\displaystyle\frac{dy}{dx}\) for \(\sqrt{xy}=1+x^2y\)

AP Calculus Exam: Multiple Choice Question

Find \(\displaystyle\frac{dy}{dx}\) for \(y=x^2e^x\).

Find the equation of the line tangent to the curve \(y=2xe^x\) at \((0,0)\).

Find \(f'(x)\) for \(f(x)=xe^x\csc{x}\).

Find \(\displaystyle\frac{dy}{dx}\) for \(e^{\frac{x}{y}}=x-y\).

Differentiate each function.

1. \(y=e^{\cos x}\)

2. \(y=\tan{(\sin x)}\)

3. \(y=\ln{(\sin t)}\)

4. \(y=e^{(1+3x)^2}\)

5. \(y=\ln{(e^{2t})}\)

6. \(z=\tan{(e^{-3\theta})}\)

Differentiate each function.

1. \(y=x^5\)

2. \(y=e^x\)

3. \(y=5^x\)

4. \(y=x^x\)

5. \(y=x^{\sin{x}}\)

6. \(y=\left(\ln{x}\right)^x\)