Derivatives and Differentiability (3.6-3.9)

What does the chain rule state?

You find the derivative by multiplying the outside of the function by the inside of it.

If you get an answer like 5^3x × 3, what should you not do?

You should not combine it to equal 15^3x.

What is the other way to find out the derivative using the chain rule?

dy/dx = dy/du × du/dx

What is the derivative of √(x²+1)?

(1/(2√(x²+1))) × 2x

How do you rewrite sin⁴(x³) to correctly find the derivative? (to make the exponent the outside)

(sin(x³))⁴

How would you rewrite y = √(x²+5x-8) to get an exponent to find the derivative?

(x²+5x-8)^1/2

How do you find cot⁻¹ (x) on the calculator?

(π/2) - tan⁻¹ (x)

What does -cos²θ - sin²θ =?

-1

What is the derivative of cscx?

-cscxcotx

What is the derivative of cotx?

-csc²x

What is the derivative of cosx?

-sinx

What is the derivative of 0?

0

What do you do when you are solving for dy/dx and there is nothing in front of the dy/dx for the part?

1. Divide the other part of the equation by the stuff being multiplied by the dy/dx 2. Complete other steps to get just the dy/dx by itself

How do you find the equations of the tangent and normal lines for an equation like x² + y² = 169?

1. Find the derivative (something equal to 0) 2. Solve for dy/dx 3. You will have to plug in both the x and y-values into the derivative to get the slope at that point

How do you find the derivative of an equation like x³ + y³ = 18xy?

1. Find the derivative of first plus derivative of second. 2. Get all dy/dx's on one side, all other parts on the other 3. Factor out the dy/dx 4. Divide the other side by stuff that was being multiplied by dy/dx

What do you do when you are given the point (2,1) and asked that given a small square what is f'(2) and f''(2)?

1. Find the derivative of the equation 2. Plug in the x and y-values 3. Find the second derivative 4. Plug in answer from first derivative to get answer to second derivative

How do you find points on a curve where the tangent is parallel to the x-axis, and parallel to the y-axis?

1. Find the derivative of the equation 2. Solve for either the x or y-value 3. Plug that into the original equation to solve for the other value 4. Plug the solved value into the equation for the other value 5. Create two different points, points parallel to x-axis go in their original order 6. For the points parallel to the y-axis, you switch the x and y-values for the points parallel to the x-axis

If given two equations like y = u⁴ and u = 5x-2, how do you find dy/dx?

1. Find the derivative of the first equation (4u³), and multiply it by the derivative of the second equation (5) 2. Or you could plug the values of u into the y equation and then find the derivative

What do you do when asked to find ( f o g)' at the given value of x and you are given f(u) = 1/cos²u, u = g(x) = πx, x = 1/4

1. Find the derivative of the first function f(u). 2. Find the derivative of the second equation (u). 3. Multiply them together 4. Set the u equation equal to the x, solve for u 5. Plug the value of u in to the multiplied derivatives to get your answer

What do you do when you have to find the derivative of an equation twice using implicit differentiation?

1. Find the first derivative 2. Find the second 3. Plug in the first derivative in for the dy/dx's 4. Simplify

How do you solve for the derivative when you have a smaller fraction in a bigger fraction?

1. Multiply by the smaller denominator over the smaller denominator 2. If there are two different denominators then multiply by first denominator by second denominator over first denominator by second denominator

If you are given an equation like f(x) = x⁵ + 2x³ + x - 1 and are asked to find f(1), f'(1), f⁻¹(3), and (f⁻¹)'(3) what do you do?

1. Plug 1 into the original f(x) equation to get the y-value 2. Find the derivative of the equation and then plug in 1 into f(x) to find the slope at that point 3. To find the points for the f⁻¹ graph you just switch the x and y-values from the f(x) equation 4. To find the slope of the f⁻¹ graph you just do the reciprocal of the f(x) graph

How do you confirm that a point (-1,1) is a on a curve defined by x³y² = cos(πy)?

1. Plug in the x and y-values 2. Check and see if both sides are equal

How do you verify that there is more than one possible y-value when x=2 for y³-xy = -1?

1. Plug in to for x 2. Set the equation equal to 0 3. Graph that equation (x³-2x+1) 4. Find x-intercepts

How do you find an inverse?

1. Switch the x and y 2. Solve for y

What do you do when asked to find where the slope of an equation is defined like x = cosy?

1. Take the derivative of both sides 2. Solve for dy/dx 3. Find where it can be defined (-1/siny), defined everywhere except where siny=0

What do you have to do when you have a variable in the base and in the exponent?

1. Take the ln of both sides 2. Bring the exponents to the front 3. Product rule on the left 4. Multiply it by the substituted y

What do you do when you are solving equations with large values with exponents?

1. Take the ln of both sides 2. If they are multiplying you can make it the ln of something plus the ln of something 3. If they are dividing you can make it the ln of something minus the ln of something 4. Move the exponents to the front 5. Multiply by y

How do you distribute (2x)²?

4x²

If you have something like √(u²) on the bottom of a equation what do you simplify it down to?

Absolute value of u

How do you find the derivative of something like ln(1/x)?

Change it to ln(x⁻¹)

What do you do when you have something like 1+cos(y)/sin(2x) = 4 cos(x)?

Cross multiply

If you are finding the derivative of something and have to use the chain rule to find it and you take the derivative of the trigonometric function and get two trigonometric functions, what is your next step? Ex. dy/dx cscx = -cscxcotx

Find the derivative of the inside functions. Since it is the same thing inside for both functions you only have to do it once.

When taking the derivative of something like 9tan(x/3) what do you do with the 9?

Keep the nine multiplied by sec²(x/3)

If you have something like log base 9 of (6x⁴+3)⁵ or ln (6x⁴+3)⁵ in your original equation, what can you do?

Move the exponent to the front.

How would you combine something like (-s²/√(1-s²)) + (1-s²)^1/2 - (1/√(1-s²))?

Multiply the middle term by ((1-s²)^1/2)/((1-s²)^1/2).

What is the normal line of the equation when the tangent is equal to something like y = 1/e that goes through the point (1, 1/e)?

Normal: x = 1

How do you find the derivative of something like y = xe^-x?

Product Rule

What is something you should do when you end up having to take the ln of both sides and multiplying one side by Y?

Put brackets around the rest of the equation being multiplied by y.

When they say "differentiate with respect to x", what are they asking you to really do?

Take the derivative of both sides of the equal sign.

If the slope of the f(x)'s tangent line at (a,b) is m, then what does that mean about the inverse graph?

The slope of the f⁻¹(x)'s tangent line at (b,a) is 1/m, reciprocal, not opposite reciprocal

What is the derivative of t/4?

This equals t × 1/4, so the derivative is just 1/4.

What do you make sure you do when finding the derivative of something like tan⁻¹(t²)?

When you are plugging the t² into the equation, make sure you do the power by the power.

What is the derivative of sinx?

cosx

What does 1/200 sec²(y) equal?

cos²y/200

How do you find sec⁻¹ (x) on the calculator?

cos⁻¹ (1/x)

If you are finding the derivative of a letter other than x, what do you multiply by?

d(other letter)/dx

What is the equation for the derivative of cos⁻¹(u)?

d/dx (cos⁻¹(u)) = (-1/√(1-u²)) × du/dx

What is the equation for the derivative of cot⁻¹(u)?

d/dx (cot⁻¹(u)) = (-1/(1+u²)) × du/dx

What is the equation for the derivative of csc⁻¹(u)?

d/dx (csc⁻¹(u)) = (-1/(|u|√(u²-1)) × du/dx

How do you take the derivative of a ln with a variable? (ln u)

d/dx (ln u) = 1/u × du/dx

What is the equation for the derivative of sec⁻¹(u)?

d/dx (sec⁻¹(u)) = (1/(|u|√(u²-1)) × du/dx

What is the equation for the derivative of sin⁻¹(u)?

d/dx (sin⁻¹(u)) = (1/√(1-u²)) × du/dx

What is the equation for the derivative of tan⁻¹(u)?

d/dx (tan⁻¹(u)) = (1/(1+u²)) × du/dx

How do you take the derivative of a number with a variable in its exponent? (a^u)

d/dx(a^u) = a^u × lna × du/dx

How do you take the derivative of e with variable in its exponent? (e^u)

d/dx(e^u) = e^u × du/dx

How do you take the derivative of a log with a number as its base and then a variable? (log base a of u)

d/dx(log base a of u) = (1/(u × lna)) × du/dx

What is the derivative of something like xe²?

e²

What is the derivative of secx?

secxtanx

What is the derivative of tanx?

sec²x

How do you find csc⁻¹ (x) on the calculator?

sin⁻¹ (1/x)

f(f⁻¹(x)) = ?

x

f⁻¹(f(x)) = ?

x

What line are inverses symmetric over?

y = x