AP Calculus Unit 3 Differentiation - Composite, Implicit, and Inverse Functions

University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2022
Views:

323

Pages:

2
Author:

ddifenwadbyx

Unit 3: Differentiation - Composite, Implicit, and Inverse Functions ● Chain Rule ○ 1) write out the outside function and the inside function ■ Ex. G(f(x)) ● Inside function: f(x) ; Outside function: G(I) ○ 2) write down the derivative of those ■ Inside function: f(x) = f’(x) O: G(I) = G’(I) ○ 3) Plug in the original inside equation to the derivative of the outside equation ■ G’(f(x)) ○ Then multiply by the derivative of the inside function ■ G’(f(x)) * f’(x) ● Implicit Differentiation Equations: equations with x and y values on the same side of the equal sign. ○ 1) With respect to x, take the derivative of the equation ■ For Y variables, take the derivative normally and multiply it by dy/dx ○ 2) Use algebra to get all dy/dx on to one side and x values onto the other ○ 3) Factor out the dy/dx from one side ○ 4) Divide by the remaining numbers so dy/dx is isolated ● Second Derivative with Implicit Equations o 1) Isolate dy/dx (what we did above) o 2) Take the second derivative of the dy/dx equation o 3) Plug in the equation to the dy/dx in the function ● Derivatives of Inverse Functions o Inverse Function: A function that switches the input and output. o To take an inverse, switch the x and y of the original function, and then manipulate it so you get it back to the general form (y = mx + b) o Inverse Function Theorem: If f is differentiable at ‘a’ and f(a) ≠ 0, then f is an open interval containing b=f(a) and the derivative of f -1 (b) is: o Derivative of Inverse trig functions: -1

AP Calculus Unit 3 Differentiation - Composite, Implicit, and Inverse Functions

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