Textbook Question
7–28. Derivatives Evaluate the following derivatives.
d/dx (ln (cos² x))
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6. Derivatives of Inverse, Exponential, & Logarithmic Functions
Derivatives of Exponential & Logarithmic Functions
1:10 minutesProblem 7.1.14
Textbook Question
7–28. Derivatives Evaluate the following derivatives.
d/dx (x^{π})
Verified step by step guidance1
Step 1: Recognize that the given function is x raised to the power of π, where π is a constant. The general formula for differentiating x raised to a constant power is d/dx(x^c) = c * x^(c-1), where c is a constant.
Step 2: Identify the constant c in this problem. Here, c = π, which is a mathematical constant approximately equal to 3.14159.
Step 3: Apply the differentiation formula. Substitute c = π into the formula, resulting in d/dx(x^π) = π * x^(π-1).
Step 4: Simplify the expression. The derivative is now expressed as π * x^(π-1).
Step 5: Conclude that the derivative has been successfully computed symbolically. No further simplification is needed unless numerical evaluation is required.
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope at any given point.
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Power Rule
The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation for polynomial and power functions, making it easier to compute derivatives quickly.
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Constant Exponents
In calculus, when dealing with functions that have constant exponents, such as x raised to π (a constant), the Power Rule still applies. The exponent π is treated as a constant, allowing us to differentiate the function using the same principles as with integer exponents. This concept is crucial for understanding how to handle derivatives of functions involving irrational or non-integer exponents.
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