BC Formulas (1)

Basic Differentiation Rules

1. Constant Rule:

   • cu: ( \frac{d}{dx}(cu) = c\frac{du'}{dx} )

2. Sum/Difference Rule:

   • [u ± v]: ( \frac{d}{dx}(u ± v) = u' ± v' )

3. Product Rule:

   • [uv]: ( \frac{d}{dx}(uv) = u'v + uv' )

4. Quotient Rule:

   • [v/u]: ( \frac{d}{dx}(\frac{u}{v}) = \frac{u'v - uv'}{v^2} )

5. Power Rule:

   • [u^n]: ( \frac{d}{dx}(u^n) = nu^{n-1}u' )

6. Exponential Rule:

   • [e^u]: ( \frac{d}{dx}(e^u) = e^u u' )

7. Logarithmic Rule:

   • [ln(u)]: ( \frac{d}{dx}(\ln(u)) = \frac{u'}{u} )

8. Trigonometric Functions:

   • Sine: ( \frac{d}{dx}(\sin(u)) = \cos(u)u' )

   • Cosine: ( \frac{d}{dx}(\cos(u)) = -\sin(u)u' )

   • Tangent: ( \frac{d}{dx}(\tan(u)) = \sec^2(u)u' )

   • Cotangent: ( \frac{d}{dx}(\cot(u)) = -\csc^2(u)u' )

   • Secant: ( \frac{d}{dx}(\sec(u)) = \sec(u)\tan(u)u' )

   • Cosecant: ( \frac{d}{dx}(\csc(u)) = -\csc(u)\cot(u)u' )

   • Arcsine: ( \frac{d}{dx}(\arcsin(u)) = \frac{u'}{\sqrt{1-u^2}} )

   • Arccosine: ( \frac{d}{dx}(\arccos(u)) = -\frac{u'}{\sqrt{1-u^2}} )

   • Arctangent: ( \frac{d}{dx}(\arctan(u)) = \frac{u'}{1+u^2} )

   • Arccotangent: ( \frac{d}{dx}(\arccot(u)) = -\frac{u'}{1+u^2} )

   • Arcsecant: ( \frac{d}{dx}(\arcsec(u)) = \frac{u'}{|u|\sqrt{u^2-1}} )

   • Arccosecant: ( \frac{d}{dx}(\arccsc(u)) = -\frac{u'}{|u|\sqrt{u^2-1}} )

Basic Integration Formulas

1. Scalar Multiplication:

   • ( \int k f(u) du = k \int f(u) du )

2. Sum/Difference Rule:

   • ( \int [f(u) ± g(u)] du = \int f(u) du ± \int g(u) du )

3. Power Rule For Integrals:

   • ( \int u^n du = \frac{u^{n+1}}{n+1} + C ) (where n ≠ -1)

4. Exponential Integral:

   • ( \int e^u du = e^u + C )

5. Trigonometric Integrals:

   • ( \int \sin(u) du = -\cos(u) + C )

   • ( \int \cos(u) du = \sin(u) + C )

   • ( \int \tan(u) du = -\ln|\cos(u)| + C )

   • ( \int \sec^2(u) du = \tan(u) + C )

   • ( \int \csc^2(u) du = -\cot(u) + C )

   • ( \int \sec(u)\tan(u) du = \sec(u) + C )

   • ( \int \csc(u)\cot(u) du = -\csc(u) + C )

   • ( \int \sin^2(u) du = \frac{1}{2}(u - \sin(u)\cos(u)) + C )

6. Inverse Trigonometric Integrals:

   • ( \int \frac{du}{a^2 + u^2} = \frac{1}{a}\arctan(\frac{u}{a}) + C )

   • ( \int \frac{du}{|u|\sqrt{u^2-1}} = \arcsec(|u|) + C )

   • ( \int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin(\frac{u}{a}) + C )