Textbook Question
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 2x/(x² + 1)³
6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
2:47 minutesProblem 3.R.7
Textbook Question
Use differentiation to verify each equation.
d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).
Verified step by step guidance1
Start by differentiating the first term, x⁴. The derivative of x⁴ with respect to x is 4x³. This is because the power rule states that d/dx(xⁿ) = nxⁿ⁻¹.
Next, differentiate the second term, ln(x⁴ + 1). Use the chain rule here: the derivative of ln(u) is 1/u times the derivative of u. Let u = x⁴ + 1, so the derivative of u with respect to x is 4x³.
Apply the chain rule: the derivative of ln(x⁴ + 1) is (1/(x⁴ + 1)) * 4x³.
Combine the derivatives: the derivative of the entire expression x⁴ − ln(x⁴ + 1) is 4x³ - (4x³/(x⁴ + 1)).
Simplify the expression: factor out 4x³ from the terms to get 4x³(1 - 1/(x⁴ + 1)). Simplify further to obtain 4x⁷/(1 + x⁴), which matches the given equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable, providing insights into the function's behavior, such as its slope at any given point. Techniques for differentiation include the power rule, product rule, quotient rule, and chain rule, each applicable in different scenarios.
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Chain Rule
The chain rule is a specific differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function. This rule is essential for differentiating functions where one function is nested within another, allowing for the correct application of derivatives in complex expressions.
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Logarithmic Differentiation
Logarithmic differentiation is a method used to differentiate functions that are products or quotients of variables raised to powers, especially when they involve logarithmic functions. By taking the natural logarithm of both sides of an equation, it simplifies the differentiation process, particularly when dealing with exponential growth or decay. This technique is particularly useful for functions that are difficult to differentiate using standard rules.
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