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Find $ f'(x) $ and $ f"(x). $

$ f(x) = (x^3 + 1)e^e $

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$=e^{x}\left(x^{3}+6 x^{2}+6 x+1\right)$

01:19

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Oregon State University

Idaho State University

Boston College

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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it's Claire's, the one you right here. So we're given up of X is equal to x cubed plus one turns e to the X We're gonna differentiate by applying the product rule and we get X cubed plus one d over t x Eat the X plus b to the x d over d x X cute plus one and we're gonna apply to some rule And when we simplify, we get e to the X times X cubed plus three X square close one. Now we're gonna determine second derivative So we're just gonna take the derivative of the 1st 1 We're just gonna ply the product and some rule once again and we get X cubed plus three x square plus one D over d x her e to the X plus e to the x d over d X execute plus three X square plus one. This becomes equal to X cubed less three x square plus one comes Eat the x close Eat the x three X square plus six sex because is equal to X cubed plus six x square plus six x plus one tongues heat, fax

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