Calculus

Calculus

This is a course that Henry and Ben are studying during 2021.

Differentiation

Power Rule

$(x^{n})'=n*x^{n-1}$

Derivative of Polynomial Functions

1. The sum rule in Newton notation is $(f+g)'=f'+g'$
2. The sum rule in Leibniz notation is ${d(f+g) \over dx}={df \over dx}+{dg \over dx}$

1. The Difference Rule 1 $(f-g)'=f'-g'$
2. The Difference Rule 2 ${d(f-g) \over dx}={df \over dx}-{dg \over dx}$

1. The Product Rule 1$(f*g)'=f*g'+g*f'$
2. The Product Rule 2${d(fg) \over dx}'=f{dg \over dx}+g{df \over dx}$

1. The Quotient Rule 1$({f \over g})'={(gf'-fg') \over g^{2}}$

1. The Quotient Rule 2${d({f \over g}) \over dx}={g{df \over dx}-f{dg \over dx} \over g^{2}}$

Newton Derivative of Polynomial Functions

1. The sum rule $(f+g)'=f'+g'$
2. The Difference Rule $(f-g)'=f'-g'$
3. The Product Rule $(f*g)'=f*g'+g*f'$
4. The Quotient Rule $({f \over g})'={(gf'-fg') \over g^{2}}$

Leibniz Derivative of Polynomial Functions

1. The sum rule ${d(f+g) \over dx}={df \over dx}+{dg \over dx}$
2. The Difference Rule ${d(f-g) \over dx}={df \over dx}-{dg \over dx}$
3. The Product Rule ${d(fg) \over dx}'=f{dg \over dx}+g{df \over dx}$
4. The Quotient Rule ${d({f \over g}) \over dx}={g{df \over dx}-f{dg \over dx} \over g^{2}}$

Derivative of Trigonometric Functions

1. $(sinx)'=cos(x)$
2. $(cosx)'=-sin(x)$
3. $(tanx)'=sec^{2}(x)$
4. $(cotx)'=-csc^{2}(x)$
5. $(cscx)'=-csc(x)cot(x)$
6. $(secx)'=sec(x)tan(x)$

Chain Rule

$[f(g(x))]'=f'(g(x))*g'(x)$

${\frac {df(g(x))}{dx}}={{df(g)} \over dg}{dg \over dx}$