Biomedical engineers in Kenya frequently encounter problems requiring precise rate of change analysis, such as modelling physiological signals, analysing biomedical instrumentation outputs, or optimizing drug delivery rates. Differentiation provides the mathematical tools to extract instantaneous rates and sensitivities from complex biomedical functions. Mastering differentiation techniques is essential for accurate interpretation and design in biomedical engineering applications.

1.1 Definition of derivatives of a function

Differentiation measures how a function changes as its input changes, giving the instantaneous rate of change or slope at any point. The derivative of a function \(f(x)\) at a point \(x\) is defined as the limit of the average rate of change as the interval approaches zero.

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

Worked Examples

Example 1: A biomedical signal \(f(t) = 4t^2 + 3t\) represents voltage output over time. Find the rate of change of voltage at \(t = 2\) seconds.

Given: \(f(t) = 4t^2 + 3t\), \(t = 2\) s

$$ f'(t) = \frac{d}{dt} (4t^2 + 3t) $$

$$ = 8t + 3 $$

At \(t=2\),

$$ f'(2) = 8(2) + 3 = 16 + 3 = 19 $$

Answer: 19 \text{ volts/second}

Example 2: The concentration \(C(t)\) of a drug in blood is \(C(t) = 5e^{-0.2t}\). Find the rate of change at \(t = 3\) hours.

Given: \(C(t) = 5e^{-0.2t}\), \(t=3\) h

$$ C'(t) = \frac{d}{dt} 5e^{-0.2t} = 5 \times (-0.2) e^{-0.2t} = -1 e^{-0.2t} $$

At \(t=3\),

$$ C'(3) = -1 \times e^{-0.6} = -1 \times 0.5488 = -0.5488 $$

Answer: -0.5488 \text{ mg/L per hour}

Example 3: The displacement \(s(t) = t^3 - 6t^2 + 9t\) models a biomechanical joint over time. Find the velocity at \(t=4\) seconds.

Given: \(s(t) = t^3 - 6t^2 + 9t\), \(t=4\) s

$$ s'(t) = 3t^2 - 12t + 9 $$

At \(t=4\),

$$ s'(4) = 3(16) - 12(4) + 9 = 48 - 48 + 9 = 9 $$

Answer: 9 \text{ m/s}

Chapter Summary

This chapter introduced the concept of derivatives as a fundamental tool for understanding how functions change. Differentiation was first explored from the first principle, establishing the foundational limit definition. A selection of common derivatives was presented in tabular form to facilitate quick reference. The chapter then detailed the main rules of differentiation, including the sum and difference, product, quotient, and chain rules, each essential for handling different functional forms. Techniques for parametric and implicit differentiation expanded the application scope to more complex relationships. Finally, the chapter examined how derivatives describe the rate of change and approximate small changes in quantities, linking the theory to practical engineering contexts. Throughout, the focus remained on developing skills to differentiate functions accurately and apply these results effectively.