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:

• Differentiation using the power rule

• Derivative notation ($f^{\prime }(x)$ and $\frac{dy}{dx}$)

• Basic differentiation rules (constant, constant multiple, sum)

• Finding slopes of tangent lines

• Evaluating derivatives at specific points

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Important Formulas

Below are the key formulas used in this exercise:

| Constant Rule | $\frac{d}{dx}(c)=0$ |

| Constant Multiple Rule | $\frac{d}{dx}[cf(x)]=c{f}^{\prime }(x)$ |

| Sum/Difference Rule | $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime }(x)\pm g^{\prime }(x)$ | | Tangent Slope at $x=a$ | $m=f^{\prime }(a)={\frac{dy}{dx}\Vert }_{x=a}$ |

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Summary

This exercise covers fundamental differentiation techniques for computing derivatives of algebraic functions. Key learnings include applying the power rule to determine $f^{\prime }(x)$ and $\frac{dy}{dx}$, combining basic differentiation rules for polynomial expressions, and evaluating derivatives at specific points to calculate the slope of tangent lines. Common strategies involve simplifying expressions before applying differentiation rules and carefully handling negative and fractional exponents.