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Chapter 15: Problem 2

If \(f\) is a function of \(x\), write down two ways in which the derivative can be written.

Short Answer

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Question: Provide two ways of representing the derivative of a function f(x) with respect to x. Answer: Two ways to represent the derivative of a function f(x) with respect to x are: 1. Using Leibniz notation: \(\frac{d}{dx}f(x)\) 2. Using prime notation (Lagrange notation): \(f'(x)\)

Step by step solution

01

Method 1: Leibniz notation

One way to write the derivative of a function f(x) with respect to x is using Leibniz notation. In this notation, the derivative is represented as: \(\frac{d}{dx}f(x)\) where the expression \(\frac{d}{dx}\) stands for taking the derivative with respect to x.
02

Method 2: Prime notation

Another way to write the derivative of a function f(x) with respect to x is using prime notation, also known as Lagrange notation. In this notation, the derivative is represented as: \(f'(x)\) where the prime symbol (') indicates taking the derivative with respect to x.

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Most popular questions from this chapter

Calculate \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) where \(y\) is given by (a) \(3 x^{4}-2 x+\ln x\) (b) \(\sin 5 x-5 \cos x\) (c) \((x+1)^{2}\) (d) \(\mathrm{e}^{3 x}+2 \mathrm{e}^{-2 x}+1\) (e) \(5+5 x+\frac{5}{x}+5 \ln x\)

The function \(y(x)\) is given by \(y(x)=x^{3}-3 x\). Calculate the intervals on which \(y\) is (a) increasing, (b) decreasing.

Find the third and fourth derivatives of \(y\) given the second derivative of \(y\) is (a) \(\frac{2}{\mathrm{e}^{3 x}}\) (b) \(\frac{1+x}{x^{2}}\) (c) \(3 \ln x^{2}\) (d) \(\sin x+\sin (-2 x)\) (e) \(\frac{\cos ^{2} x+\cos x}{\cos x}\)

Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) where \(y(x)\) is defined by (a) \(3 x^{2}-\mathrm{e}^{2 x}\) (b) \(\sin 3 x+\cos x\) (c) \(\sqrt{x}\) (d) \(\mathrm{e}^{x}+\mathrm{e}^{-x}\) (e) \(1+x+x^{2}+\ln x\)

If \(x\) is a function of the independent variable \(t\), write down two ways in which the derivative can be written.
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