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

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

Calculus

6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Derivatives of Exponential & Logarithmic Functions

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1:59 minutes

Problem 7.1.21

Textbook Question

7–28. Derivatives Evaluate the following derivatives.

d/dx (e^{-10x²})

Verified step by step guidance

Step 1: Recognize that the function involves the exponential function e raised to a power, specifically e^{-10x²}. The derivative of e^u with respect to x is e^u * du/dx, where u is a function of x.

Step 2: Identify the inner function u = -10x². To apply the chain rule, we need to compute the derivative of u with respect to x.

Step 3: Differentiate u = -10x² with respect to x. Using the power rule, the derivative of x² is 2x, and multiplying by the constant -10 gives du/dx = -20x.

Step 4: Apply the chain rule. The derivative of e^{-10x²} with respect to x is e^{-10x²} * (-20x), combining the derivative of the outer function e^u and the inner function u = -10x².

Step 5: Simplify the expression. The derivative is -20x * e^{-10x²}. This is the simplified form of the derivative.

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Key Concepts

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

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. Understanding how to compute derivatives is essential for analyzing the behavior of functions, including their maxima, minima, and points of inflection.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of the natural logarithm, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay, depending on the sign of 'b'. In the context of derivatives, the derivative of an exponential function is proportional to the function itself, which simplifies the differentiation process.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, 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 particularly useful when dealing with functions that involve exponentials, as seen in the differentiation of e^{-10x²}.

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