Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
y = x⁵
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3. Techniques of Differentiation
Basic Rules of Differentiation
2:31 minutesProblem 3.25
Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
h(t) = t²/2 + 1
Verified step by step guidance1
Step 1: Identify the function for which you need to find the derivative. Here, the function is \( h(t) = \frac{t^2}{2} + 1 \).
Step 2: Apply the power rule for derivatives. The power rule states that if \( f(t) = t^n \), then \( f'(t) = n \cdot t^{n-1} \).
Step 3: Differentiate the first term \( \frac{t^2}{2} \). Using the power rule, the derivative of \( t^2 \) is \( 2t \). Since it is divided by 2, the derivative becomes \( \frac{2t}{2} = t \).
Step 4: Differentiate the constant term \( 1 \). The derivative of a constant is always 0.
Step 5: Combine the derivatives of each term to find the derivative of the entire function. Thus, \( h'(t) = t + 0 = t \).
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2mKey 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 provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule.
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Derivatives
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, especially for polynomial functions, making it easier to compute derivatives quickly.
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Function Notation
Function notation is a way to represent functions in mathematics, typically denoted as f(x) or g(t). In the context of the question, h(t) = t²/2 + 1 is a function of t, and understanding this notation is crucial for identifying how to apply differentiation techniques. It helps clarify the relationship between the input variable and the output of the function.
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