Main rules:

\[ \frac{d}{dx}(x^n)=nx^{n-1} \]

\[ \frac{d}{dx}(c)=0 \]

\[ \frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x)) \]

If

\[ y=uv, \]

then

If

\[ y=\frac{u}{v}, \]

then

If

\[ y=f(g(x)), \]

then

\[ \frac{d}{dx}(e^x)=e^x \]

\[ \frac{d}{dx}\bigl(e^{f(x)}\bigr)=f'(x)e^{f(x)} \]

\[ \frac{d}{dx}(a^x)=a^x\ln a \]

\[ \frac{d}{dx}\bigl(a^{f(x)}\bigr)=f'(x)a^{f(x)}\ln a \]

\[ \frac{d}{dx}(\ln x)=\frac{1}{x} \]

\[ \frac{d}{dx}\bigl(\ln(f(x))\bigr)=\frac{f'(x)}{f(x)} \]

\[ \frac{d}{dx}(\sin x)=\cos x \]

\[ \frac{d}{dx}(\cos x)=-\sin x \]

\[ \frac{d}{dx}(\tan x)=\sec^2 x \]

With the chain rule:

\[ \frac{d}{dx}(\sin(f(x)))=f'(x)\cos(f(x)) \]

\[ \frac{d}{dx}(\cos(f(x)))=-f'(x)\sin(f(x)) \]

\[ \frac{d}{dx}(\tan(f(x)))=f'(x)\sec^2(f(x)) \]

When differentiating \(y\)-terms:

\[ \frac{d}{dx}(y)=\frac{dy}{dx} \]

\[ \frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx} \]

If

\[ x=f(t), \qquad y=g(t), \]

then

\[ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]

\[ \frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}} \]

Equation of tangent:

\[ y-y_1=m(x-x_1) \]

Gradient of normal:

\[ m_{\text{normal}}=-\frac{1}{m_{\text{tangent}}} \]