Question

Differentiate each of the following: a. $y=e^{3 x}$ b. $s=e^{3 t-5}$ c. $y=2 e^{10 t}$ d. $y=e^{-3 x}$ e. $y=e^{5-6 x+x^{2}}$ f. $y=e^{\sqrt{x}}$

Name: Problem 2, Chapter 5: Nelson Calculus and Vectors
Uploaded: 2021-12-12T21:11:29-08:00
Duration: 3 min 4 s
Channel: Gregory Higby
Description: Differentiate each of the following: a. y=e^3 x b. s=e^3 t-5 c. y=2 e^10 t d. y=e^-3 x e. y=e^5-6 x+x^2 f. y=e^√(x)

   Differentiate each of the following:
a. $y=e^{3 x}$
b. $s=e^{3 t-5}$
c. $y=2 e^{10 t}$
d. $y=e^{-3 x}$
e. $y=e^{5-6 x+x^{2}}$
f. $y=e^{\sqrt{x}}$

Nelson Calculus and Vectors

Chris Kirkpatrick,… 12th Edition

Chapter 5, Problem 2

Instant Answer

Solved by Expert Gregory Higby

12/12/2021

Step 1

This is a direct application of the chain rule. Show more…

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Differentiate each of the following: a. $y=e^{3 x}$ b. $s=e^{3 t-5}$ c. $y=2 e^{10 t}$ d. $y=e^{-3 x}$ e. $y=e^{5-6 x+x^{2}}$ f. $y=e^{\sqrt{x}}$

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Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. When a function is composed of an outer function and an inner function, the derivative of the composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Exponential Function Differentiation

Exponential function differentiation involves taking the derivative of functions in the form y = e^(g(x)). The process applies the chain rule, resulting in the derivative y' = e^(g(x)) * g'(x), which accounts for both the exponential part and the derivative of the exponent.

Constant Multiple Rule

The constant multiple rule states that when a function is multiplied by a constant, the derivative of the product is simply the constant multiplied by the derivative of the function. This simplifies the process of differentiating functions that include constant coefficients.

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