Session 08 - Chain Rule and Derivatives of Exponential, Logarithmic, and Inverse Trig Functions

Engineering Context: Multilayer Systems and Feedback Loops

The calculus principles introduced in these guided notes are essential for engineering systems that involve dynamic layering or translate linear motion into rotational and exponential states. The Chain Rule combined with exponential functions allows electrical and aerospace engineers to calculate transient current surges in discharging circuits ($V(t) = V_0 e^{-t/RC}$) and predict aerodynamic stress thresholds (Max-Q) as vehicles traverse rapidly changing atmospheric densities. Furthermore, logarithmic derivatives govern the optimization of thrust-to-weight ratios in rocket propulsion via the Tsiolkovsky rocket equation ($\Delta v = v_e \ln(m_0/m_f)$), while inverse trigonometric derivatives are utilized in robotics and radar telemetry to translate linear tracking coordinates into precise angular motor velocities ($\frac{d\theta}{dt}$) using the derivative of $\tan^{-1}(u)$.

• Review common misconceptions and clarify key notations.

• Reflect on the physical modeling applications and mathematical setups.

• Verify calculations and mathematical reasoning.

• Core takeaways from Session 08 and overview of homework Knewton: The Chain Rule Derivatives of Exponential Functions Derivatives of Log and Exponential Functions.