Derivatives
cos(x/2)
+or- Square root (1+cos(x))/2
sin(x/2)
+or- square root (1-cos(x))/2
d/dx (csc(x))
-csc(x)cot(x)
d/dx (cot(x))
-csc^2(x)
d/dx (cos(x))
-sin(x)
lim (theta-->0) (cos(theta)-1)/(theta)
0
lim (theta-->0) (sin(theta))/(theta)
1
Steps for the chain rule
1. Let u = g(x) and find u' 2. rewrite function in terms of u 3. derivative = f'(u) . u' 4. substitute back for u to original value
sec(x) =
1/cos(x)
csc(x) =
1/sin(x)
sin(2x)
2sin(x)cos(x)
Quotient Rule
If f and g are differentiable at x and if g(x) doesn't equal 0, then so is the quotient f/g differentiable at x and d/dx(f(x)/g(x)) = (g(x)d/dxf(x)-f(x)d/dxg(x))/(g(x))^2
Theorem 1 - differentiability implies continuity
If f has a derivative at x=c then f is continuous at x=c. The converse of this is false.
Tangent Line Definition
The tangent line y=f(x) at the point P(a, f(a)) to the curve y=f(x) is the line through P with the slope m=lim(x-->a) (f(x)-f(a))/(x-a)
Difference rule
a combination of the sum and constant rule
Sum Rule
at every point where both f and g are differentiable d/dx (f(x) + g(x)) = d/dx f(x) + d/dx g(x)
d/dx (sin(x))
cos(x)
cot(x) =
cos(x)/sin(x)
cos(x+y)
cos(x)cos(y)-sin(x)sin(y)
cos(2x)
cos^2(x)-sin^2(x)=1-2sin^2(x)=2cos^2(x)-1
derivative of the natural exponential function
d/dx e^x =e^x
Product Rule
if f and g are differentiable at x, then so is their product fg, and d/dx (f(x) x g(x)) = f(x) d/dx g(x) - g(x) d/dx f(x)
the chain rule
if f(u) is differentiable at the point u=g(x) and g(x) is differentiable at x, then the composite function F(x)=(f o g) = f(g(x)) is differentiable at x, and F'(x) = f'(g(x) . g'(x)
Constant multiple rule
if f(x) is a differentiable function, and c is a constant, then d/dx (c x f(x)) = c x f'(x) = c x d/dx f(x)
Derivative of a constant function
if f(x)=c for a constant c then d/dx (c) = 0, and d/d(x) (x) =1
Power Rule
if n is any real number, then d/dx (x^n) = nx^n-1 for all x where the powers x^n, x^n-1 are defined.
alternative equation of tangent line
m= lim (h-->0) (f(x+h)-f(x))/h
d/dx (sec(x))
sec(x)tan(x)
d/dx (tan(x))
sec^2(x)
tan(x) =
sin(x)/cos(x)
sin(x+y)=
sin(x)cos(y)+sin(y)cos(x)
equation of the tangent line
y-y0=m(x-x0)
