Section 1
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Last updated
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Date created
Mar 1, 2020
Cards (51)
Section 1
(50 cards)
sech x=
1/ cosh x
The product rule
d/dx[f(x)•g(x)]= f(x) • d/dx[g(x)] + g(x) • d/dx [f(x)] or u • v' + v • u'
The constant multiple rule
If c is a constant and f is a differentiable function, the d/dx[cf(x)]= c•d/dx f(x)
The sum rule
If f and g are both differentiable, then d/dx[f(x) + g(x)]= d/dx f(x) + d/dx g(x)
Implicit differentiation
consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y'
d/dx ln | x |
1/x
d/dx (cot x)
-csc^2 x
differential
dy= f'(x)dx dx= change in x
coth x=
cosh x/ sinh x
d/dx (csc x)
-csc x cot x
cosh^2(x)-sinh^2(x)
1
The only solutions of the differential equation dy/dt = ky are the exponential functions
y(t)= y(0)e^kt
d/dx (log b x)
1/ x ln b
law of natural growth
dy/dt= ky if k>0
d/dx (tanh x)
sech^2 x
csch x=
1/ sinh x
derivative of the natural exponential function
d/dx (e^x)= e^x
d/dx (cos x)
-sin x
steps in logarithmic differentiation
1. take natural logarithms of both sides of an equation y=f(x) and use the laws of logarithms to simplify 2. differentiate respectfully to x 3. solve the resulting equation for y'
d/dx (csch x)
-csch x coth x
Horizontal tangents
occur where the derivative is 0
sinh(-x)=
-sinh x
d/dx (cos^-1 x)
- 1/sqrt (1-x^2)
the chain rule
if g is differentiable at x and f is differentiable at g(x), then the composite function F = f • g defined by F(x)= f(g(x)) is differentiable at x and F' is given by the product F'(x)= f'(g(x)) • g'(x)
1-tanh^2(x)
sech^2(x)
sinh(x+y)=
sinh(x)cosh(y)+cosh(x)sinh(y)
d/dx (sec x)
sec x tan x
d/dx (sin^-1 x)
1/ sqrt (1-x^2)
The difference rule
If f and g are both differentiable, then d/dx[f(x)-g(x)]= d/dx f(x) - d/dx g(x)
The power rule
If n is any real number, then d/dx(x^n) = nx^(n-1)
d/dx (sinh x)=
cosh x
The quotient rule
d/dx [f(x)/g(x)]= g(x) • d/dx[f(x)] - f(x) • d/dx [g(x)] / [g(x)^2] or v • u' - u • v' / v^2
d/dx (sin x)
cos x
cosh(x+y)=
cosh(x)cosh(y)+sinh(x)sinh(y)
d/dx (csc^-1 x)
-1/x sqrt (x^2-1)
cosh x=
(e^x + e^-x)/ 2
law of natural decay
dy/dt =ky if k<0
d/dx (ln x)
1/x
Derivative of a constant function
d/dx(c)= 0
d/dx (cosh x)
sinh x
d/dx (cot^-1 x)
- 1/ 1 + x^2
linear approximation formula (linearization)
L(x)= f(a)+f'(a)(x-a)
sinh x=
(e^x - e^-x)/ 2
d/dx (sec^-1 x)
1/ x sqrt(x^2-1)
d/dx tan^-1 x
1/1 + x^2
d/dx (tan x)
sec^2 x
cosh(-x)=
cosh x
d/dx (sech x)
-sech x tanh x
the power rule combined with the chain rule
if n is any real number and u= g(x) is differentiable, then d/dx [g(x)]^n = n[g(x)]^(n-1) • g'(x)
tanh x=
sinh x/ cosh x
Section 2
(1 card)