Derivative Rules

Mean Value Theorem (MVT)

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists an x=c such that f'(c)=[(f(b)-f(a))/(b-a)]

Linearization

L(x)=f(a) + f'(a)(x-a)

Rolle's Theorem

Let f(x) be continuous on [a,b] and differentiable on (a,b). If f(a)=f(b), then there exists some x=c such that f'(c)=0

Position, Velocity, and Acceleration

Position Function: s(t) Velocity Function: s'(t)=v(t) Acceleration Function: s''(t)=v'(t)=a(t)

The Quotient Rule

The derivative of (u/v)=[v(du/dx) - u(dv/dx)]÷v² [(lo d high) - (high d lo)]÷lo²

The Constant Rule

The derivative of a constant is 0

The Constant Multiple Rule

The derivative of c(f(x))=c(f'(x))

The Sum and Difference Rule

The derivative of f(x)+g(x)=f'(x)+g'(x)

The Product Rule

The derivative of uv=[u(dv/dx)] + [v(du/dx)]

The Power Rule

The derivative of xⁿ=n(xⁿ⁻¹)

Finding Points of Inflection

The points of inflection on the original function's graph occur where the first derivative's graph has max's/mins, and where the second derivative is =0. If the graph is concave up, then the tangent lines lie below the graph. If the graph is concave down, then the tangent lines lie above the graph.

d/dx [|u|]=

[(u)(u')]/(|u|) , u≠0

d/dx [|x-1|]=

[(x-1)(x-1)']/(|x-1|) = (x-1)/(|x-1|)

d/dx [sin(x)]=

cos(x)

What is the notation of the derivative of a function y?

dy/dx

d/dx [x∧e]=

e(x∧(e-1))

d/dx [e∧u]=

e∧u × u'

d/dx [e∧x]=

e∧x

Find f'(x) if f(x)=(e∧x)/((e∧x)-1)

f'(x)= -(e∧x)/((e∧x)-1)²

Find f'(x) if f(x)=√(x) - (1/√(x))

f'(x)=1/(2√(x)) + 1/(2x√(x))

Find f'(x) if f(x)=4x³-3x²+2x-π

f'(x)=12x²-6x+2

Differentials

f(a+dx)≈f(a) + dy

d/dx [uⁿ]=

n(uⁿ⁻¹)(u')

d/dx [sec(x)]=

sec(x) × tan(x)

d/dx [tan(x)]=

sec²(x)

d/dx [arctan(u)]=

u'/(1+u²)

d/dx [arcsec(u)]=

u'/(|u|√(u²-1))

d/dx [arcsin(u)]=

u'/(√(1-u²))

Find y' if y=6x∧(-3/2) + 7x∧(1/5) +1

y'= -9x∧(-5/2) + (7/5)x∧(-4/5)

Find y' if y=cos³(√(x))

y'= −[(3cos²√(x)(sin(x))]/2√(x)

Find y' if y=(e∧(2x))(sin(3x))

y'=(2e∧(2x))sin3x+(3e∧(2x))cos3x

Find y' if y=arcsin(x²)

y'=(2x)/(√(1-x⁴))

The equation of the tangent line to the curve y=f(x) at the point (a, f(a)) is given by what formula?

y-f(a) = f'(a) (x-a)

d/dx [log∨a(u)]=

(1/(ln(a))u)u' = u'/(ln(a))u

d/dx [ln(u)]=

(1/u)u' = u'/u

d/dx [a∧u]=

(ln(a))(a∧u)u'

d/dx [a∧x]=

(ln(a))(a∧x)

Find y' if y=√(sin(x))

y'=(cos(x))/[2√(sin(x))]

d/dx [csc(x)]=

-csc(x) × cot(x)

d/dx [cot(x)]=

-csc²(x)

d/dx [cos(x])=

-sin(x)

d/dx [arccot(u)]=

-u'/(1+u²)

d/dx [arccsc(u)]=

-u'/(|u|√(u²-1))

d/dx [arccos(u)]=

-u'/(√(1-u²))

d/dx [e∧e]=

0

How to find the equation of a horizontal tangent on a function

1) Take a derivative 2) Set derivative = 0 3) Solve for the variable 4) Write it as: y=(what you solved for the variable)

How to find the equation of a vertical tangent on a function

1) Take a derivative 2) Set derivative's denominator = 0 3) Solve for the variable 4) Write it as x=(what you solved for the variable)

Implicit Differentiation Process

1. Differentiate both sides of the equation with respect to x, treating y as a function of x (Chain Rule). 2. Collect the terms with "dy/dx" on one side of the equation, and those without it on the other. 3. Factor out "dy/dx" (if necessary). 4. Solve for "dy/dx".

Steps for Related Rates Problems

1. Draw and label a picture 2. Use appropriate variables to represent values 3. Label all rates of change given and those to be determined, using calculus notation (dV/dt, dr/dt, etc.) 4. Write an equation to relate variables. 5. Substitute any CONSTANT values. 6. Differentiate (by implicit differentiation) with respect to t. 7. Substitute all known values and evaluate. 8. Don't forget your units!

Find y' if y=sin(2x)/cos(2x)

y'=2sec²(2x)

Find y' if y=tan(6x)

y'=6sec²(6x)

How to find the max and min of f on the interval [a,b]

1. Make sure f is continuous on [a,b] 2. Find the critical numbers of f(x) in (a,b). This is where the derivative=0 or is undefined. 3. Evaluate f(x) at each critical number in (a,b) 4. Evaluate f(x) at each endpoint in [a,b] 5. The least of these values is the minimum. The greatest is the maximum.

Optimization

1. Sketch a picture and label given values 2. Determine the Primary Equation - This is for the value that is to be determined (maximized or minimized). 3. Determine the Secondary Equation - Used to reduce primary equation to one variable. 4. Determine domain of the Primary Equation. 5. Minimize/Maximize over CLOSED interval using the first derivative and endpoints.

Logarithmic Differentiation Process

1. Take the log of both sides. 2. Apply the properties of logarithms. 3. Differentiate the equation. 4. Solve for "dy/dx". 5. Use the original function to substitute for "y".