Derivatives Flashcards
Slope
Rise over run
y/x
Derivate (∂) (d/dx)
Slope of the tangent line as it changes with (x)
∂=f’(x)= Lim h→0 (f(x+h)-f(x))/h
Average rate of change
m=(f(x)-f(a))/(x-a)
Aka: secant line
Secant line
Average rate of change between two points
m=(f(x)-f(a))/(x-a)
Differentiable if…
Continuous, no other criteria
Derivative of any constant
Zero
Derivative of a variable to a power (x^n)
Exponent times base to the power of (exponent-1)
nx^(n-1)
Derivative of a Function multiplied by a constant
Equal to the same constant multiplied by the function derivative
∂[cf(x)]=cf’(x)
Derivative of two functions added/subtracted together
Equal to the derivatives of those functions added/subtracted together
∂[f(x)±g(x)]=f’(x)±g’(x)
Derivative of e^x
Goes unchanged
∂e^x = e^x
Second derivative [f”(x)]
∂[f’(x)]=f”(x)
Any-order derivative formula
[[∂f(x)]^n]/[∂(x^n)]
Formula to the n-th derivative
Derivative of the product of two functions
∂[f(x)*g(x)]
Sum of the products with the derivative switching places
f’(x)g(x)+f(x)g’(x)
Derivative of the quotient of two functions
∂[f(x)/g(x)]
Difference of the products with the derivative switching places, over second function squared
[f’(x)g(x)-f(x)g’(x)]/[g(x)^2]
Derivative of e^(kx)
Unchanged but multiplied by k
k*e^(kx)
Derivative sin(x)
Cos(x)
Trigonometric derivative chain
Sin(x)→Cos(x)→-Sin(x)→-Cos(x)→repeat
Derivative -sin(x)
-Cos(x)
Derivative cos(x)
-sin(x)
Derivative -cos(x)
Sin(x)
Derivative tan(x)
Sec^2 (x)
Derivative -tan(x)
-sec^2 (x)
Derivative -cot(x)
Csc^2 (x)
Derivative cot(x)
-csc^2 (x)
Differentiation process
1) chain rule first, always
2) turn exponent to nx^(n-1)
3) factor out the constant and e^x multipliers
4) use product rule, use quotient rule
5) use sum/difference rule
6) replace trig functions with their variables
Derivative sec(x)
Sec(x)*tan(x)
Derivative -sec(x)
-Sec(x)*tan(x)
Derrivative csc(x)
-csc(x)*cot(x)
Derrivative -csc(x)
csc(x)*cot(x)
Instantaneous values
Lim (x-xi)→0= (f(x)-f(xi))/(x-xi)
When (x+xi)=[desiredValue]
Average cost
(C(x)-C(xi))/(x-xi)
C(x)- cost of producing x-items
Marginal cost
The approximate cost of producing one more item after youyoure first x items
C’(x)
Chain rule- for (f o g)
Derivative of the first function, of interior function, times derivative of interior function of contents
f(g(x))=f’(g(x))*g’(x)
Derivative for a function to a power
∂[f(x)^n]=f'(x)*n(f(x))^(n-1) Derivative f(x) times exponent rule
Implicit differentiation (dy/dx)
1) Define y as y(x)
2) Take the derivative of each term
3) Rearrange to from y(x)y’(x)
3) Turn y(x)y’(x) into ydy/dx
4) Isolate the term containing ydy/dx
5) Reduce to its simplest form
Derivative of ln x
1/x
Derivative of a constant raised to a variable (b^x)
(b^x)*ln(b)
Derivative log-b x
1/[x*ln(x)]
Derivative of sin^-1(x)
1/√(1-x^2)
Derivative of tan^-1(x)
1/(1+x^2)
Derivative cos^-1(x)
-1/√(1-x^2)
Derivative cot^-1(x)
-1/(1+x^2)
Derivative sec^-1(x)
1/(|x|*√(x^2-1))
Derivative csc^-1(x)
-1/(|x|*√(x^2-1))
Steps for related rate problems
1) Write equations that express basic relationships between variables
2) Introduce rates of change by differentiating the appropriate equations with respect to time
3) Introduce rates of change by differentiating the approprite equations with respect to time
4) Substitute known values and solve for the desired quantity
5) check that the untis are reasonable
Absolute maximum
The greatest output on an entire curve
Absolute Minimum
The least output on an entire curve
Extreme Value Theorem
On a closed interval [a,b], the curve has both a minimum and a maximum value
Local minimum
The least possible output on the interval [a,b]
Local maximum
The greatest possible output on the interval [a,b]
Extreme point theorem
The derivative of a maxima or a minima is always zero
∂[extrema]=0
The derivative of a maxima or a minima
Is always zero
Finding extrema values
1) Solve for the derivative
2) Set equal to zero
3) Simplify
4) Isolate x
5) Use algebra until you find a set value
Concave up
Positive Derivative
Curves up, approaching infinity
Visualize an upward opening parabola
Concave down
Negative derivative
Curves down
Visualize a downward opening parabola
Inflection point
Any point at which the derivative goes from + to -, or from - to +
Always f”(x)=0
Second derivative test
When f’(x)=0
If f”(x)>0 → minimum
If f”(x)<0 → maximum
Objective function
Quantity you wish to maximize
Maximizing objective functions
1) Write the functions that you know
2) Eliminate all but one of the independent variables via substitution
3) Use algebra to convert this to an algebraic function
4) Calculate the derivative
5) set equal to zero
6) Solve for x
Linear approximation
Use the output of the line tangent to a nearby point to approximate the actual function output at that value, f(a) is the tangent line
f(x)≈f(a)+f’(x)(x-a)
Differntials
Functions that describe variations between the line tangent to a nearby point and the actual function output at that value, f(a) is the tangent line
Δy=f(a+Δx)-f(a)
Mean Value Theorem (Rolle’s Theorem)
On every interval [a,b] there is a value ‘c’ between ‘a’ and ‘b’, equal to the average slope on that interval
f’(c)=[f(b)-f(a)]/(b-a)
Lhopital’s rule
Any limit f(x)/g(x)= same limit f’(x)/g’(x)
True if f(x) and g(x) limits are 0/0 ∞/∞ 0*∞ ∞-∞ 1^∞ 0^0 ∞^0
Growth rates
f(x) grows faster than g(x) if
Lim→∞ g(x)/f(x)=0
Or
Lim→∞ f(x)/g(x)=∞
The rates of growth (dy/dt)
Growth Rates=∂[Pe^(rt)]=Pre^(rt)=rA(t)
Described as dy/dt
Rate constant (k)
The rate by which Pe^(rt) grows exponentially
Here, it is the ‘r’
Relative growth rate
Rate divided by current output
(dy/dt)/y
Always equal to ‘k’ (or ‘r’)
Doubling time
Time it takes to before the initial value doubles
T2=ln2/k
Exponential decay
Describes how P decreases with time
Takes the form: P-e^(rt)
Halflife
Time it takes for the decay function to reach half its original value
T(1/2)=ln(2)/k
Economic Elasticity
D
Atomic Kinetics
S
Newton’s Methods
S
Oscilators
S
Partial derivatives
1) Pick the variable indicated in the problem
2) calculate the derivative as if all the other variables were actually constants
Newtons Notation/Lagrange’s Notation
Marks derivatives as
F’(x)
Leibniz’s Notation
Mark derivarves as
d/dx
Slop of the Tangent line
Instantaneous Rate of change for the curve,
Slope at a point
Lim (x-xi)→0 for (f(x)-f(xi))/(x-xi)
So long as x=[point in question]
