1. The problem is to know and use the derivatives of standard functions: $x^n$ (for any rational $n$), $\sin x$, $\cos x$, $\tan x$, $e^x$, and $\ln x$, with examples. 2. Derivative of $x^n$ (power rule): $$\frac{d}{dx} x^n = n x^{n-1}$$ Example: $\frac{d}{dx} x^3 = 3x^2$ 3. Derivative of $\sin x$: $$\frac{d}{dx} \sin x = \cos x$$ Example: $\frac{d}{dx} \sin x = \cos x$ 4. Derivative of $\cos x$: $$\frac{d}{dx} \cos x = -\sin x$$ Example: $\frac{d}{dx} \cos x = -\sin x$ 5. Derivative of $\tan x$: $$\frac{d}{dx} \tan x = \sec^2 x$$ Example: $\frac{d}{dx} \tan x = \sec^2 x$ 6. Derivative of $e^x$: $$\frac{d}{dx} e^x = e^x$$ Example: $\frac{d}{dx} e^x = e^x$ 7. Derivative of $\ln x$ (for $x>0$): $$\frac{d}{dx} \ln x = \frac{1}{x}$$ Example: $\frac{d}{dx} \ln x = \frac{1}{x}$ 8. Summary: Use these rules to differentiate functions involving these standard forms. Final answer: The derivatives are $\frac{d}{dx} x^n = n x^{n-1}$, $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$, $\frac{d}{dx} \tan x = \sec^2 x$, $\frac{d}{dx} e^x = e^x$, and $\frac{d}{dx} \ln x = \frac{1}{x}$.