Differential and Integral Calculus

Chapter 4 of by Feliciano and Uy focuses on the Differentiation of Transcendental Functions . This chapter expands beyond algebraic rules to cover trigonometric, exponential, logarithmic, and hyperbolic functions. Core Topics and Objectives

• Theorem: If $y = f(u)$ and $u = g(x)$, then the derivative of $y$ with respect to $x$ is: $$ \fracdydx = \fracdydu \cdot \fracdudx $$
• The "Onion Analogy": The text often suggests differentiating from the "outside in." One differentiates the outer function first, keeping the inner function intact, and then multiplies by the derivative of the inner function.

C. Fencing/Cost problems

Definition:

It is used for finding the derivative of composite functions (a function within a function).

). Feliciano and Uy emphasize the pattern: the first times the derivative of the second, plus the second times the derivative of the first.

• Theorem: If $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$.
• Justification: Geometrically, a constant function is a horizontal line, which has a slope of zero.

Example (from Feliciano & Uy, typical problem):

Find the equations of tangent and normal to (y = x^2 - 4x + 3) at (x = 2). Solution :

Key Takeaways