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Chapter 5: Problem 37

In Exercises \(33-54,\) find the derivative of the function. $$y=e^{x-4}$$

Short Answer

Expert verified
The derivative of the function \(y=e^{x-4}\) is \(y'=e^{x-4}\)

Step by step solution

01

Identify the Outer and Inner Functions

In the function \(y=e^{x-4}\), the outer function is the exponential function represented by 'e', and the inner function is \(x-4\)
02

Apply the Chain Rule

Apply the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. For the outer function 'e' the derivative is itself and for the inner function \(x-4\), the derivative is 1.
03

Compute the Derivative

By chain rule, the derivative of \(y=e^{x-4}\) is its derivative with respect to the inner function, multiplied by the derivative of the inner function. Hence the derivative is \(e^{x-4} * 1 = e^{x-4}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule
The chain rule is an essential method for differentiating composite functions, which involve one function inside another. It helps break down complex derivatives into manageable parts. Imagine you have a function that consists of two layers: an outer function and an inner function. To apply the chain rule, you:
  • Differentiate the outer function with respect to the inner function, keeping the inner function unchanged.
  • Multiply the result by the derivative of the inner function.
Using the chain rule allows you to tackle derivatives of complex functions step by step by treating them as a composition of simpler functions. When applied correctly, the chain rule enables you to simplify differentiation tasks into easier computations.
Exponential Function
The exponential function, particularly with the base 'e', is an interesting and unique function in calculus. This function is expressed as \(e^x\), where 'e' is an irrational constant approximately equal to 2.71828. The crucial aspect of this function is its derivative property:
  • The derivative of \(e^x\) is itself; \(e^x\).
This property makes the exponential function remarkably easy to differentiate compared to polynomials or trigonometric functions. Exponential functions are prevalent in natural growth processes, such as population growth, radioactive decay, and certain financial models. Their self-replicating derivative characteristic keeps them central to many calculus operations.
Composite Function
A composite function arises when one function is nested inside another. This is often depicted as \(f(g(x))\), where 'f' is the outer function and 'g' is the inner function. Composite functions allow you to describe more complex relationships by combining simpler functions.
  • The inner function \(g(x)\) can be any simple function like \(x-4\).
  • The outer function \(f(u)\) can be more complex like \(e^u\).
To differentiate a composite function, using the chain rule is crucial as it effectively handles the layers within the function. Through recognizing the composite nature, you ensure accurate differentiation, helping to maintain precision in calculus problems.

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Most popular questions from this chapter

Describing a Graph Use a graphing utility to graph the function \(f(x)=\arccos x+\arcsin x\) on the interval \([-1,1] .\) (a) Describe the graph of \(f\) . (b) Verify the result of part (a) analytically.

Given \(e^{x} \geq 1\) for \(x \geq 0,\) it follows that $$ \int_{0}^{x} e^{t} d t \geq \int_{0}^{x} 1 d t $$ $$ \begin{array}{l} e^{x} \geq 1+x \\ \text { for } x \geq 0 \end{array} $$

Using Properties of Exponents In Exercises \(107-110\) , find the exact value of the expression. $$6^{(\ln 10) / \ln 6}$$

Verifying Identities In Exercises 81 and \(82,\) verify each identity. (a) \(\operatorname{arccsc} x=\arcsin \frac{1}{x}, \quad|x| \geq 1\) (b) \(\arctan x+\arctan \frac{1}{x}=\frac{\pi}{2}, \quad x>0\)

An Approximation of e Complete the table to demonstrate that \(e\) can also be defined as \(\lim _{x \rightarrow 0^{+}}(1+x)^{1 / x}\)
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