AP Calculus BC Unit 2: Differentiation: Definition and Fundamental Properties

Find $\frac{d}{dx}\left[\sqrt{x}\cdot e^x\right]$.

If $f$ is differentiable at $x=a$, then ${\lim }_{x\to a}\frac{f(x)-f(a)}{x-a}$ equals $f^{\prime }(a)$. Using this definition, find $\frac{d}{dx}[x^{-1}]$ at $x=2$.

What is $\frac{d}{dx}[\ln (x^3)]$ for $x>0$?

Using the limit definition, find $f^{\prime }(x)$ for $f(x)=3x^2-5$.

Find $\frac{d}{dx}\left[\frac{x^2+1}{x-3}\right]$ at $x=4$.

Find the equation of the tangent line to $y=x\ln x$ at the point $(e,e)$.

Find $\frac{d}{dx}\left[x^3\ln x\right]$ for $x>0$.

The derivative of $f(x)=\frac{x^2+1}{x-3}$ at $x=4$ is:

Find $\frac{d}{dx}\left[\tan x\right]$ directly from $\sin x/\cos x$ using the quotient rule.

Let $f(x)=x^x$ for $x>0$. Find $f^{\prime }(x)$.

Let $f(x)=x^{4/3}-2x^{1/3}$. Find all $x$ where $f^{\prime }(x)=0$.

If $f(x)=e^{2x}\cos x$, find $f^{\prime }(x)$.