Calculus Project - Derivatives!
Find sec(5pi/6).
(-2(sqrt3))/3
Differentiate (4x - 9)^(3/5)
(12/5)(4x - 9)^(-2/5)
Find the average velocity during the first 5 seconds.
(202 - -2)/(5-0) Answer = 40.8
Differentiate y= sin (xcosx)
(cosx - xsinx) cos(xcosx)
Differentiate log(5^x)
(ln(5))/(ln(10))
Find tan(pi/6)
(sqrt 3)/3
Differentiate cot(8x)
-8csc²(8x)
What is the derivative of cotx?
-csc^2x
What is the derivative of cscx?
-cscxcotx
Find y' of cscx - cosx
-cscxcotx + sinx
What is the derivative of cosx?
-sinx
Find y' of (cosx)/(tanx - sinx)
1. ((-sinx)(tanx-sinx) - (sec^2x - cosx)(cosx))/(tanx-sinx)^2 2. Simplify: ((-sinxtanx + sin^2x) - (sec^2xcosx - cos^2))/(tanx-sinx)^2 Answer: -((sinxtanx -( 1 + secx))/(tanx-sinx)^2
Find y' of x^2cosx
1. 2x(cosx) + x^2(-sinx) Simplify Answer: 2xcosx - x^2sinx
Where does v(t) = 0?
1. 3t^2 + 8t - 5 = 0 2. Use the quadratic formula to find the 0's 3. v = 0 @ v= .522, -3.189
Find a(4).
1. 6t + 8 2. 6(4) + 8 3. Answer: 32
What are the rules for differentiation?
1. The Constant Rule: d/dx(c)= 0 2. Power Rule: d/dx(x^n)=nx^n-1
What are the Pythagorean identities?
1. cos^2x + sin^2x = 1 2. cot^2x + 1 = csc^2x 3. 1 + tan^2x = sec^2x 4. (cos^2)/sin^2x + sin^2x/sin^2x = 1/sin^2x 5. cos^2x/cos^2x + sin^2x/cos^2x = 1/cos^2x
Find the change in direction of s(t) = t^2 - 4t - 5
1. s'(t) = 2t -4 2. 2t-4 = 0 3. 2t = 4 4. t = 2
Let's do a problem. s(t)=t^3 + 4t^2 -5t + 2 Find v(2)
1. s'(t) = 3t^2 + 8t - 5 2. v(3) = 3(2)^2 + 8(2) - 5 3. Answer = 23
Find the derivative of secx.
1. secx = 1/cosx 2. y' = ((0(cosx) -1(-sinx))/cos^2x 3. sinx/cosx(cosx) 4. sinx/cosx x 1/cosx 5. tanxsecx
Find the body's velocity when the acceleration is 0.
1. v(t) = 3t^2 + 8t - 5 2. a(t)= 6t + 8 = 0 3. a(t) = 0 @ t=-1 1/3 4. 3(-1 1/3)^2 + 8(-1 1/3) - 5 5. Answer = -10 1/3
Differentiate y=arcsin(x−1)
1/√2x−x^2
Differentiate x²+y²−2x−4y=4
1−x/y−2.
Find the derivative of ln^2(10x)
2ln(10x)/x
Differentiate y=arccot1x2
2x/1+x^4
Differentiate dy/dx=3x²/2y
2x³=2y²+5
Solve (6x + 5)^3
3(6x + 5)^2 x 6 Answer: 18(6x + 5)^2
Differentiate 1/(9-5x)⁶
30/(9-5x)⁷
Differentiate sec^2 x + tan^2 x
4 sec^2 x tanx
Find the displacement of the body after 5 seconds.
5^3 + 4(5)^2 - 5(5) + 2 = 202 0^3 +4(0)^2 - 5(0) - 2 = -2 202 - -2 = 204 = Answer
Solve: y=x^5+3x^4+2x^2+5 using the Power Rule.
5x^4 + 12x^3 + 4x
Write an equation relating surface area of a cube with its side length.
A = 6s^2
But what is a tangent line?
A tangent line is a line that touches the initial line at a certain point, with its slope being that of the derivative. It is 'parallel' to the graph at that point.
Find the derivative of the above answer.
A' = 12s
Evaluate A'(5) and A'(6)
A'(5) = 12(5) = 60 A'(6) = 12(6) = 72
Using the rules of differentiation, solve x^3 + 3x^2 +4x +5
Answer: 3x^2 + 6x + 4
What is the Power Rule?
Basically, a fast version of the Constant Multiple Rule, with a little variation. Example Below:
How do we find acceleration?
By taking the derivative of velocity or the second derivative of position.
Does differentiability imply continuity?
If f has a derivative at x=a, then f is continuous at x=a.
What is acceleration?
It is the rate at which a object's velocity changes(increase or decrease) It also measure how fast the object increases or decreases in speed.
What does it mean to be differentiable?
It means that you can take the derivative at a certain point.
Does continuity imply differentiability?
No, just because f(x) is continuous does not mean it is also differentiable. Like the graph of the abs(x), it is continuous, but not differentiable at x=0.
Example of Product Rule: Solve d/dx (x-2)(x^2 + 5)
Step 1. (x-2) = u, (x^2 + 5) = v Step 2. (x-2)(2x) + (1)(x^2 + 5) Step 3. Distribute: (2x^2 -4x) + (x^2 +5) Step 3. Add together common terms or cancel out. Step 4. Answer: 3x^2 - 4x + 5
Solve (4x^2)(2x+5) using the Product Rule.
Step 1. 4x^2 = u, 2x+5=v Step 2. (8x(2x+5)) + ((2)(4x^2)) Step 3. Distribute: (16x^2+40x) + 8x^2 Step 4. Answer: 24x^2 + 40x
Using the Constant Multiple Rule, solve y=7x^2
Step 1. Dy/dx 7x^2 = 7(dy/dx x^2) Step 2. 7(2x) Step 3. Answer: 14x
How do we use the definition of the derivative to find the slope?
Step 1. We have to plug in the f(x) equation into the definition
Using the Constant Multiple rule, solve y= 5/x^4
Step 1. dy/dx(5x^-4) = 5(dy/dx x^-4) Step 2. 5(-4x^-5) Step 3. Answer: -20/x^5
Based on the equation f(x) = 7x^3, find the slope at x=4
Step 1. f'(x) = 7(3x^2) = 21x^2 Step 2. f'(2) = 21(4)^2 = 21(16) = 336
Take f'''''(x) of y= 7x^4. (the 5th derivative)
Step 1. f'(x)= 28x^3 step 2. F''(x)=84x^2 Step 3. f'''(x)=168x Step 4. f''''(x)=168 Step 5. f'''''(x)= 0
Use the quotient rule to solve y= (x^2 + 5x)/(x^3)
Step 1. x^2 + 5x=u, v= x^3 Step 2. ((x^3)(2x+5) - 3x^2(x^2 + 5x))/(x^3)^2 Step 3. Distribute: ((2x^4 + 5x^3) - (3x^4 + 15x^3))/x^6 Step 4. Combine Terms Answer: (-x^4 -10x^3)/x^6
Example: Use the Quotient Rule to solve y= (x^4)/(x-2)
Step 1. x^4 = u, x-2=v Step 2. ((x-2)(4x^3)) - (x^4)(1))/(x-2)^2 Step 3. Distribute: ((4x^4 -8x^3) - (x^4))/(x-2)^2 Step 4. Cancel or combine terms: (3x^4 - 8x^3)/(x-2)^2
Step 2?
Step 2. We then distribute throughout the equation, plugging the x value in f(x).
Step 3?
Step 3. If needed, cancel out what can be, including variable if you need to factor.
Implicit Differentiation
Steps: 1. Differentiate both sides with respect to x 2. Get all terms with dy/dx to one side of the equation 3. Factor out dy/dx 4. Solve for dy/dx
how do we find speed?
Taking the absolute value of the velocity equation.
What does velocity = 0 signify?
That the object has stopped or changed direction.
How do we find displacement?
The change in position, the endpoint - the start point.
What is the definition of the derivative?
The definition of the derivative finds the slope of the line tangent at a certain point.
Where do we get the derivative?
The derivative is taken from the original function, f(x) or such.
What is a derivative?
The derivative is the equation that finds the slope to a line tangent to the graph.
How do we get the derivative?
The first step for new students is the definition of the derivative.
Step 4?
Then, plug in the value in which h approaches, and you will have your derivative!
How many types of non-differentiablity?
There are 4!
What does velocity tell us?
Velocity tells us the direction of motion in which an object is moving.
What is the first one?
Where there is a corner on the graph of f(x)!
What is the second one?
Where there is a cusp on the graph of f(x)!
Where is the third one?
Wherever the slope is undefined/or a vertical tangent on the graph of f(x)!
Where is the fourth, and last one?
Wherever there is discontinuity, or a 'jump' on the graph of f(x)!
Is there any other way to tell if f(x) is differentiable?
Yes, if the right and left hand derivatives are not equal. They must also be continuous!
Can we take multiple derivatives of the same equation?
Yes, you can take as many as you like! For example: f''(x), f'''(x), f''''(x), etc.
Solve y=sin10x
cos10x(10) Answer: 10cos10x
What is the derivative of sinx?
cosx
Find y' of sinx + tanx
cosx + sec^2x
What is the quotient rule?
d/dx (u)/v = (vu' - uv')/v^2
What is the Product Rule?
d/dx (uv) = uv' + vu'
Derivative of e^x
d/dx [e^x] = e^x
Derivative of ln x
d/dx [ln x] = 1/x
Derivative of logbx
d/dx [logbx} = 1/ x ln b
What is the constant multiple rule?
d/dx c (f(x)) = c((d/dx f(x))
d/dx a^u
d/dx[a^u] = 1/(ulna) x du/dx
Differentiate x²-5xy+3y²=7
dy/dx= (-2x+5y)/( -5x+6y)
Differentiate 2x²-4xy+2y²=10
dy/dx=1
Find the derivative of e^tan2x
e^tan2x(sec^2(2x))(2)
What is the Chain Rule?
f'(g(x))g'(x), a way to differentiate functions within functions
Differentiate (x^2+3)^7
f'(x) = -14x(x^2+3)^6
What is position?
s(t) = position at time(t)
What is the derivative of tanx?
sec^2x
What is the derivative of secx?
secxtanx
Find csc(pi/4)
sqrt 2.
find cot(11pi/6)
sqrt 3
Find sin(pi/4)
sqrt2/2
Differentiate xtan(2x)
tan(2x) + 2xsec²(2x)
Differentiate dy/dx = -x²/y²
x³ + y³ = 8
Differentiate y=cos(x+y)
y′=−sin(x+y)/1+sin(x+y)
Find the speed @ t = 3
|3(3)^2 + 8(3) - 5| Answer: 23