Math 170

Derivative Function

d= f(x+h)-f(x)/h

(f/g) → (f/g)’

(g’ * f) - (f ’ * g)/ g²

d/dx (f - g) →

f ’ - g’

d/dx (f + g) →

f ‘ + g’

d/dx (e^x) →

e^x

xⁿ → derivative

axⁿ

= nX^n-1
= a * n X^n-1

sin x → derivative

cos x

cos x → derivative

-sin x

tan x → derivative

sec² x

csc x → derivative

- csc x cot x

sec x → derivative

+ sec x tan x

cot x → derivative

- csc² x

coefficient ( c ) → derivative

Ex. 1, 7, 5, 12

= 0

x or y→ derivative

1

d/dx (In x) → derivative

= 1/x

d/dx sin^-1x → derivative

1/√1-x²

d/dx cos^-1 x → derivative

- 1/√1-x²

d/dx tan^-1x → derivative

1/1+x²

d/dx (log_bx) → derivative

1/ x In b

s(t) = ?
s’(t) = ?
s’’(t) = ?

s(t) = position in respect to time “t”
s’ (t) = v(t) = velocity

s’’(t) = v’ (t) = acceleration

Area forumla for Circle

A = pi * r²

Circle: dA/dt = ?

(dA/ dr) * (dr/ dt)

Area formula for Rectangle

A = L * W

Rectangle: dA/dt = ?

L dw/dt + W dL/dt

Area of a triangle

a² + b² = c²

Tangent Line
Linear Approximation
How do you get the original line?

Tangent Line: y-y₁ = m (x-x₁) & m = f ’(x)
Linear Approximation: L(x) -y₁= m (x-x₁) & m = f’’ (x)

Original Line: negative reciprocal of m in tangent line

Is a graph increasing or decreasing, how do you determine?

f ‘(x) = (+) → Increasing
f ‘(x) = (-) → Decreasing

How do you determine concavity?

f’’ (+) = Concave Up

f’’ (-) = Concave Down

What is an inflection point, how is it determined?

Inflection Point: Where the graph changes from concave up to concave down (…OR VICE VERSE)

1. Calculate the first derivative, f'(x), of the original function, f(x).

Calculate the second derivative, f''(x), by differentiating f'(x).

2. Set the second derivative, f''(x), equal to zero and solve for x. 

3. Create a sign chart or test intervals around each candidate point.

Evaluate the second derivative (f''(x)) at a value within each interval.

If the sign of f''(x) changes from positive to negative, or from negative to positive, at a candidate point, then it is an inflection point.

4. Substitute the x-value(s) of the inflection point(s) into the original function, f(x), to find the corresponding y-coordinate(s).