Power Rule: \( \frac{d}{dx} \left[ x^n \right] = n x^{n-1} \)
Constant Rule: \( \frac{d}{dx} \left[ c \right] = 0 \)
Constant Multiple Rule: \( \frac{d}{dx} \left[ k f(x) \right] = k \frac{d}{dx} f(x) \)
Exponential Functions:
\( \frac{d}{dx} \left[ e^x \right] = e^x \)
\( \frac{d}{dx} \left[ e^{(ax+b)} \right] = ae^{(ax+b)} \)
\( \frac{d}{dx} \left[ a^x \right] = a^x \ln a \)
Logarithmic Functions:
\( \frac{d}{dx} \left[ \ln x) \right] = \frac{1}{x} \)
\( \frac{d}{dx} \left[ \log_a x \right] = \frac{1}{x \ln a} \)
Trigonometric Functions:
\( \frac{d}{dx} \left[ \sin(x) \right] = \cos(x) \)
\( \frac{d}{dx} \left[ \cos(x) \right] = -\sin(x) \)
\( \frac{d}{dx} \left[ \tan(x) \right] = \sec^2(x) \)
\( \frac{d}{dx} \left[ \sin(ax \pm b) \right] = a\cos(ax \pm b) \)
\( \frac{d}{dx} \left[ \csc(x) \right] = -\csc(x).\cot(x) \)
\( \frac{d}{dx} \left[ \sec(x) \right] = \sec(x).\tan(x) \)
\( \frac{d}{dx} \left[ \cot(x) \right] = -\csc^2(x) \)
Sum/Diff Rule: \( \frac{d}{dx} \left[ u \pm v \right] = \frac{du}{dx} \pm \frac{dv}{dx} \)
Product Rule: \( \frac{d}{dx} \left[ u.v \right] = u \frac{dv}{dx} + v \frac{du}{dx} \)
Quotient Rule: \( \frac{d}{dx} \left[ \frac{u}{v} \right] = \frac{v \frac{du}{dx} – u \frac{dv}{dx}}{V^2} \)
Chain Rule: \( \frac{dz}{dx} = \frac{dz}{dy} \times \frac{dy}{dx} \)
Inverse Trigonometric Functions:
\( \frac{d}{dx} \left[ \arcsin(x) \right] = \frac{1}{\sqrt{1-x^2}} \)
\( \frac{d}{dx} \left[ \arccos(x) \right] = -\frac{1}{\sqrt{1-x^2}} \)
\( \frac{d}{dx} \left[ \arctan(x) \right] = \frac{1}{1+x^2} \)
Hyperbolic Functions:
\( \frac{d}{dx} \left[ \sinh(x) \right] = \cosh(x) \)
\( \frac{d}{dx} \left[ \cosh(x) \right] = \sinh(x) \)
\( \frac{d}{dx} \left[ \tanh(x) \right] = \text{sech}^2(x) \)