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Chapter 15: Problem 1

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

Short Answer

Expert verified
Answer: The two different ways to write the derivative of a function x with respect to t are Leibniz's notation, written as (dx/dt), and prime notation, written as x'(t).

Step by step solution

01

Method 1: Leibniz's Notation

Leibniz's notation for the derivative of \(x\) with respect to \(t\) is: $$\frac{dx}{dt}$$ This notation highlights the fact that the derivative represents the ratio of the change in \(x\) to the change in \(t\) as \(t\) approaches to a particular value.
02

Method 2: Prime Notation

Prime notation, also known as Lagrange's notation, represents the derivative of a function by adding an apostrophe (prime symbol) to the function name. For the derivative of \(x\) with respect to \(t\), we would write: $$x'(t)$$ This notation is more concise and is commonly used when working with functions involving a single variable.

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Most popular questions from this chapter

Find the rate of change of the following functions: (a) \(\frac{3 t^{3}-t^{2}}{2 t}\) (b) \(\ln \sqrt{x}\) (c) \((t+2)(2 t-1)\) (d) \(\mathrm{e}^{3 v}\left(1-\mathrm{e}^{v}\right)\) (e) \(\sqrt{x}(\sqrt{x}-1)\)

Calculate \(y^{\prime \prime}(1)\) where \(y(t)\) is given by (a) \(t\left(t^{2}+1\right)\) (b) \(\sin (-2 t)\) (c) \(2 \mathrm{e}^{t}+\mathrm{e}^{2 t}\) (d) \(\frac{1}{t}\) (e) \(\cos \frac{t}{2}\)

The function \(y(x)\) is given by $$ y(x)=2 x^{3}-9 x^{2}+1 $$ (a) Calculate the values of \(x\) for which \(y^{\prime}=0\). (b) Calculate the values of \(x\) for which \(y^{\prime \prime}=0\). (c) State the interval(s) on which \(y\) is increasing. (d) State the interval(s) on which \(y\) is decreasing. (e) State the interval(s) on which \(y^{\prime}\) is increasing. (f) State the interval(s) on which \(y^{\prime}\) is decreasing.

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) \(5+5 x+\frac{5}{x}+5 \ln x\)

Evaluate the rate of change of \(H(t)=5 \sin t-3 \cos 2 t\) when (a) \(t=0\) (b) \(t=1.3\).
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