Derivatives (Advanced)

Harder items at competition-adjacent depth.

Phase 2 · AP Calculus

Derivatives: core idea

Implicit differentiation.

If $f(x)=x^x$ , find $f'(1)$ .

Hint: Log differentiation: $x^x(\ln x+1)$ . Answer: 1.

1. FTC: $\dfrac{d}{dx}\int_0^{x^2}\cos t\,dt$ at $x=1$ ?
2. Implicit: $x^2+y^2=25$ at $(3,4)$ : $dy/dx$ ?
3. If $f(x)=\ln(\ln x)$ , find $f'(e)$ .
4. If $f(x)=\arctan(x^2)$ , find $f'(1)$ .
5. Maximum of $xe^{-x}$ occurs at $x=$ ?
6. Linearization of $\sqrt{x}$ at $4$ : approximate $\sqrt{4.1}$ as $a+b/20$ with $a=2$ . Enter $b$ .
7. Where $x^3-6x^2+9x$ increases: left endpoint of increasing interval containing $3$ ?
8. If $y=x^y$ and $(x,y)=(1,1)$ , find $dy/dx$ at that point.
9. Derivative of $(\sin x)^x$ at $x=\pi/2$ ?