What is the first step in calculating derivatives for different types of functions?
Identify the function type
A composite function is formed by composing two or more functions
An implicit function has y explicitly isolated in the equation.
False
What type of function reverses the input and output of another function?
Inverse function
Match the function type with its differentiation procedure:
Composite Function ↔️ Chain Rule
Implicit Function ↔️ Implicit Differentiation
Inverse Function ↔️ Inverse Function Rule
The Chain Rule is used to differentiate composite functions
Matching the function type to the correct differentiation procedure is crucial for accuracy.
What is the formula for the Chain Rule if y = f(g(x))</latex>?
dxdy=f′(g(x))⋅g′(x)
Steps to perform implicit differentiation for x2+y2=1.
1️⃣ Differentiate both sides with respect to x
2️⃣ Apply the chain rule when differentiating y2
3️⃣ Solve for dxdy
What is the derivative of f(u)=u4?
f′(u)=4u3
The chain rule in implicit differentiation is used when differentiating y2 with respect to x
What is dxdy for the implicit function x2+y2=1?
dxdy=−yx
Match the function type with its differentiation procedure:
Composite Function ↔️ Chain Rule
Implicit Function ↔️ Implicit Differentiation
Inverse Function ↔️ Inverse Function Rule
Composite functions are differentiated using the chain rule.
Steps to differentiate a composite function using the chain rule:
1️⃣ Identify the outer and inner functions
2️⃣ Find the derivative of the outer function
3️⃣ Find the derivative of the inner function
4️⃣ Multiply the derivatives
To differentiate an implicit function, you first differentiate both sides with respect to x
What is the inverse function rule for differentiation?
dxdf−1(x)=f′(f−1(x))1
What is the key procedure in implicit differentiation?
Solve for dxdy
The inverse function rule states that \frac{d}{dx}f^{ - 1}(x) = \frac{1}{f'(f^{ - 1}(x))}</latex>, where f′(x) is the derivative of the original function
What does the chain rule state for composite functions?
dxdy=f′(g(x))⋅g′(x)
The inverse function rule is used to find the derivative of the inverse of a function.
The chain rule is used to differentiate composite functions.
Steps to apply the chain rule
1️⃣ Identify the outer function f(u) and inner function g(x)
2️⃣ Find their derivatives: f′(u) and g′(x)
3️⃣ Apply the chain rule formula
What are the outer and inner functions for y = (3x^{2} + 2x - 1)^{4}</latex>?
f(u)=u4,g(x)=3x2+2x−1
The derivative of y=(3x2+2x−1)4 is dxdy=4(3x2+2x−1)3⋅(6x+2), using the chain rule.
Implicit differentiation involves differentiating both sides of an equation with respect to x.
Steps to perform implicit differentiation
1️⃣ Differentiate both sides with respect to x
2️⃣ Apply chain rule where necessary
3️⃣ Solve for dxdy
What is the derivative dxdy for the equation x^{2} + y^{2} = 1</latex>?
dxdy=−yx
The inverse derivative formula states that dxdf−1(x)=f′(f−1(x))1, where f′(x) is the derivative of the original function.
If f(x)=x3, what is f−1(x)?
f−1(x)=3x
The derivative of f^{ - 1}(x) = \sqrt[3]{x}</latex> is dxdf−1(x)=3x321, using the inverse derivative formula.
Which two differentiation procedures are often combined for complex functions?
Chain rule and implicit differentiation
The chain rule is frequently used within implicit differentiation and the inverse function rule.
Steps to apply the chain rule for y = (3x^{2} + 2x - 1)^{4}</latex>
1️⃣ Identify the outer function f(u)=u4 and inner function g(x)=3x2+2x−1
2️⃣ Find their derivatives: f′(u)=4u3 and g′(x)=6x+2
3️⃣ Apply the chain rule formula
What is the derivative of f−1(x)=3x using the inverse derivative formula?