Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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2:16 minutes

Problem 3.45

Textbook Question

Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25

Verified step by step guidance

Step 1: Identify the function and the point at which you need to evaluate the derivative. The function is \( f(s) = 2\sqrt{s} - 1 \) and the point is \( a = 25 \).

Step 2: Rewrite the function in a form that is easier to differentiate. Express \( \sqrt{s} \) as \( s^{1/2} \). Thus, the function becomes \( f(s) = 2s^{1/2} - 1 \).

Step 3: Differentiate the function using the power rule. The power rule states that \( \frac{d}{ds}[s^n] = ns^{n-1} \). Apply this to \( 2s^{1/2} \) to find \( f'(s) \).

Step 4: Calculate the derivative \( f'(s) \). Using the power rule, \( f'(s) = 2 \cdot \frac{1}{2}s^{-1/2} = s^{-1/2} \).

Step 5: Evaluate the derivative at \( s = 25 \). Substitute \( s = 25 \) into \( f'(s) = s^{-1/2} \) to find \( f'(25) \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function's graph at that point.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function is composed of two functions, say f(g(x)), the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when dealing with functions that involve nested expressions.

Square Root Function

The square root function, denoted as √s, is a specific type of function that returns the non-negative value whose square is s. When differentiating functions involving square roots, it is important to apply the power rule appropriately, as √s can be rewritten as s^(1/2). Understanding how to manipulate and differentiate square root functions is crucial for solving problems involving them.

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Textbook Question

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.

h(x) = √x (√x-x³/²)