Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

Key Concepts

This exercise focuses on the following concepts:

• Advanced Differentiation Rules: Application of the Product Rule, Quotient Rule, and Chain Rule to complex functions.

• Implicit Differentiation: Finding the derivative $\frac{dy}{dx}$ when $y$ is not explicitly isolated.

• Parametric Differentiation: Calculating $\frac{dy}{dx}$ for curves defined by parametric equations $x=f(t)$ and $y=g(t)$.

• Differentials: Understanding the differential $dy=f^{\prime }(x)dx$ as an approximation of the change in $y$.

• Linear Approximation: Using differentials to estimate numerical values of roots, powers, and trigonometric functions.

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Important Formulas

Below are the key formulas used in this exercise:

| Quotient Rule | $\frac{d}{dx}[\frac{u}{v}]=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$ |

| Chain Rule | $\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$ |

| Implicit Derivative | $\frac{d}{dx}[f(y)]=f^{\prime }(y)\frac{dy}{dx}$ |

| Parametric Derivative | $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$ | | Differential | $dy=f^{\prime }(x)dx$ | | Linear Approximation | $f(x+\Delta x)\approx f(x)+f^{\prime }(x)\Delta x$ |

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Summary

This exercise provides a comprehensive review of differentiation techniques beyond basic power rules. It transitions from explicit differentiation (where $y$ is a function of $x$) to implicit differentiation (where $x$ and $y$ are intertwined) and parametric differentiation (where both depend on a third variable $t$).

A significant portion of the exercise is dedicated to the practical application of differentials, teaching how to use the derivative at a known point to approximate values of functions at nearby points.

The core strategy across all problems is to correctly identify the outer and inner functions to apply the Chain Rule effectively.