Calculus Formulas: Finding Derivatives

Basic Formulas of Derivatives

For any two differentiable functions f(x) and g(x):

1. f(x)=c (c is a constant), then f'(x)=0
   
2. g(x)=cf(x) (c is a constant), then g'(x)=c·f'(x)
   
3. f(x)=xⁿ (n is a rational number), then f'(x)=nx^n-1
   
4. y=f(x)±g(x), then y'=f'(x)±g'(x)
   
5. y=f(x)·g(x) then, y'=f'(x)·g(x)+f(x)·g'(x)
   
6. y=f(x)/g(x) (g(x)≠0) then, y'=(f'(x)g(x)-f(x)g'(x))/g²(x)
   

Derivatives of Various Functions

1. y=e^x then, y'=e^x
   
2. y=a^x then, y'=a^x·ln(a)
   
3. y=ln(x) then, y'=1/x
   
4. y=log_ax then, y'=1/(x·ln(a))
   
5. y=sin(x) then, y'=cos(x)
   
6. y=cos(x) then, y'=-sin(x)
   
7. y=tan(x) then, y'=1/cos²(x)
   
8. y=arctan(x) then, y'=1/(x²+1)
   
9. y=arcsin(x) then, y'=(1-x²)^-0.5
   

Derivative of Composite Functions

1. y=f(u), u=g(x) then, dy/dx=dy/du · du/dx
   
2. y=f(ax+b) then, y'=af'(ax+b)
   
3. y=fⁿ(x) then, y'=n·f^n-1(x)·f'(x)