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