3: Discovering Derivatives
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Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter so that we can then explore applications of these techniques.
- 3.1: Derivatives of Polynomial Functions
- The derivative of a constant function is zero. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decreases by 1. The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative. The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
- 3.2: Differentiation Techniques - The Product and Quotient Rules
- The derivative of a constant function is zero. The derivative of a power function is a function in which the power on x becomes the coefficient of the term and the power on x in the derivative decreases by 1. The derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative. The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
- 3.3: Derivatives of Trigonometric Functions
- We can find the derivatives of sin x and cos x by using the definition of derivative and the limit formulas found earlier. With these two formulas, we can determine the derivatives of all six basic trigonometric functions.
- 3.4: Differentiation Techniques - The Chain Rule
- Key Concepts The chain rule allows us to differentiate compositions of two or more functions. It states that for \(h(x)=f(g(x)),\) \(h′(x)=f′(g(x))g′(x).\) We can use the chain rule with other rules that we have learned, and we can derive formulas for some of them. The chain rule combines with the power rule to form a new rule: If \(h(x)=(g(x))^n\),then \(h′(x)=n(g(x))^{n−1}g′(x)\).
- 3.5: Derivatives of Exponential and Hyperbolic Functions
- In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.
- 3.6: Differentiation Techniques - Implicit Differentiation
- We use implicit differentiation to find derivatives of implicitly defined functions (functions defined by equations). By using implicit differentiation, we can find the equation of a tangent line to the graph of a curve.
- 3.7: Derivatives of Logarithmic, Inverse Trigonometric, and Inverse Hyperbolic Functions
- The inverse function theorem allows us to compute derivatives of inverse functions without using the limit definition of the derivative. We can use the inverse function theorem to develop differentiation formulas for the inverse trigonometric functions.
- 3.8: Differentiation Techniques - Logarithmic Differentiation
- In this section, we explore derivatives of exponential and logarithmic functions. As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Logarithmic functions can help rescale large quantities and are particularly helpful for rewriting complicated expressions.
- 3.9: Related Rates
- If two related quantities are changing over time, the rates at which the quantities change are related. For example, if a balloon is being filled with air, both the radius of the balloon and the volume of the balloon are increasing. In this section, we consider several problems in which two or more related quantities are changing and we study how to determine the relationship between the rates of change of these quantities.
- 3.10: Linear Approximations and Differentials
- In this section, we examine another application of derivatives: the ability to approximate functions locally by linear functions. Linear functions are the easiest functions with which to work, so they provide a useful tool for approximating function values. In addition, the ideas presented in this section are generalized later in the text when we study how to approximate functions by higher-degree polynomials Introduction to Power Series and Functions.
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