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Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.4.23
Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t
Guida verificata passo dopo passo1
Step 1: Identify the function f(t) = t^{5/3} e^t as a product of two functions: u(t) = t^{5/3} and v(t) = e^t.
Step 2: Apply the product rule for derivatives, which states that if you have a function h(t) = u(t)v(t), then h'(t) = u'(t)v(t) + u(t)v'(t).
Step 3: Differentiate u(t) = t^{5/3} using the power rule. The power rule states that if u(t) = t^n, then u'(t) = n t^{n-1}. So, u'(t) = \(\frac{5}{3}\) t^{\(\frac{5}{3}\) - 1}.
Step 4: Differentiate v(t) = e^t. The derivative of e^t with respect to t is simply e^t, so v'(t) = e^t.
Step 5: Substitute u(t), u'(t), v(t), and v'(t) into the product rule formula: f'(t) = u'(t)v(t) + u(t)v'(t) = \(\frac{5}{3}\) t^{\(\frac{2}{3}\)} e^t + t^{\(\frac{5}{3}\)} e^t.
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Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Derivatives
Product Rule
The Product Rule is a formula used to find the derivative of the product of two functions. It states that if you have two functions, u(t) and v(t), the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as polynomials and exponentials.
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Exponential Functions
Exponential functions are functions of the form f(t) = a * e^(kt), where 'e' is the base of natural logarithms, and 'a' and 'k' are constants. The derivative of an exponential function is unique because it is proportional to the function itself, making it straightforward to differentiate. Understanding how to differentiate exponential functions is crucial when they are part of more complex expressions.
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Pratica correlata
Domanda del libro di testo
Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).
Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>
y = cx²; x²+2y² = k, where c and k are constants
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Domanda del libro di testo
Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1
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Domanda del libro di testo
15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)
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Domanda del libro di testo
Find y'' for the following functions.
y = ex sin x
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Domanda del libro di testo
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(3x + 1)⁴
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