AP Calculus BC Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Find $\frac{d}{dx}\left[\arctan \left(\frac{y}{x}\right)\right]$ treating $y$ as a differentiable function of $x$.

If $g(x)=f^{-1}(x)$ and $f(3)=7$, $f^{\prime }(3)=4$, find $(g{)}^{\prime }(7)$.

Let $F(x)=f(g(x))$ where $f(u)=u^3+u$ and $g(x)=\cos x$. Find $F^{\prime }(\pi /2)$.

Use implicit differentiation to find $\frac{dy}{dx}$ for $x^3+y^3=6xy$.

Find $\frac{d}{dx}[\arcsin (\sqrt{x})]$.

Find $\frac{dy}{dx}$ by implicit differentiation if $x^2+y^2=25$.

Find $\frac{d}{dx}\left[\sin (x^3)\right]$.

If $y=\arctan (2x)$, find $\frac{dy}{dx}$.

Find $\frac{d^{2}y}{d{x}^2}$ given $y=\sin x$ and then evaluate $\frac{d^{2}y}{d{x}^2}$ for $y=x\sin x$.

Use implicit differentiation to find $\frac{d^{2}y}{d{x}^2}$ for $x^2+y^2=1$ at the point $(0,1)$.

Differentiate $f(x)=e^{{\sin }^{2}x}$.

Find $\frac{d}{dx}[\sinh x]$ using the definition $\sinh x=\frac{e^x-e^{-x}}{2}$.