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(a) Sketch, by hand, the graph of the function $ f(x) = e^x, $ paying particular attention to the graph crosses the y-axes. What fact allows you to do this?(b) What types of functions are $ f(x) = e^x $ and $ g(x) = x^e? $ Compare the differentiation formulas for $ f $ and $ g. $(c) Which of the two functions in part (b) grows more rapidly when $ x $ is large?

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01:31

Amy Jiang

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 1

Derivatives of Polynomials and Exponential Functions

Derivatives

Differentiation

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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(a) Sketch, by hand, the g…

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Use the graph of $y=e^{x}$…

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$$f(x)=\left\{\begin{array…

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The graph of a function $f…

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(a) Sketch the graph of $f…

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and slower. So when you read here so we're gonna draw a graph for part A. We have X is equal to eat the axe, and when Xs equal to zero, we have a wide value of one. The derivative is equal to eat the axe, so on access equal to zero, the derivative is equal to one, so it's increasing when it cuts the Y axis. For part B, we have f of X is exponential and it's defined for all ex Brian's and it's equal to FX. G of X is equal to X to the e and it's defined for only positive values of X, the derivative of F ISS needs the axe, then the derivative of G this e times x to the A minus one for part c Well, you see that FX is equal to eat. The X grows more rapidly when x this large compared to Jesus the X because when you see the graph up of X looks like this g of x, well, look like this

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