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A function is defined as f(x) = a(e^x+a)-x. Discuss the monologue of this equati...

Mar 16, 2024

A function is defined as f(x) = a(e^x+a)-x. Discuss the monologue of this equation.

Solution by Steps

step 1

To find the derivative of the function

f(x) = a(e^x + a) - x

, we will apply the rules of differentiation

step 2

Differentiate each term of

f(x)

separately. The derivative of

a(e^x)

with respect to

x

a e^x

, using the constant multiple rule and the exponential rule

step 3

The derivative of the constant

a^2

is 0, since the derivative of any constant is 0

step 4

The derivative of

-x

with respect to

x

-1

, using the power rule

step 5

Combine the derivatives from the previous steps to get the derivative of

f(x)

f'(x) = a e^x - 1

Answer

f'(x) = a e^x - 1

Key Concept

Differentiation of Exponential Functions and Constants

Explanation

The derivative of an exponential function

e^x

is itself, and the derivative of a constant is 0. When differentiating a function that is a combination of exponential terms and constants, apply the rules of differentiation to each term separately.