Textbook Question
The following limits represent f'(a) for some function f and some real number a.
b. Evaluate the limit by computing f'(a).
lim x🠂0 e^x-1 / x
3. Techniques of Differentiation
Basic Rules of Differentiation
2:31 minutesProblem 3.28
Textbook Question
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
g(t) = 6√t
Verified step by step guidance1
Step 1: Recognize that the function g(t) = 6√t can be rewritten using exponent notation as g(t) = 6t^{1/2}.
Step 2: Apply the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n*t^{n-1}.
Step 3: Differentiate g(t) = 6t^{1/2} using the power rule. The derivative of t^{1/2} is (1/2)t^{-1/2}.
Step 4: Multiply the derivative of t^{1/2} by the constant 6, resulting in g'(t) = 6 * (1/2)t^{-1/2}.
Step 5: Simplify the expression to find the derivative g'(t) = 3t^{-1/2}.
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Key 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 its 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 chain rule.
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Power Rule
The power rule is a basic technique for finding the derivative of functions of the form f(x) = x^n, where n is a real number. According to this rule, the derivative f'(x) is given by n*x^(n-1). This rule simplifies the differentiation process, especially for polynomial functions and can be applied to functions involving roots, such as √t.
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Square Root Function
The square root function, denoted as √x, is a specific type of function that returns the non-negative square root of x. In calculus, it is important to understand how to differentiate this function, as it can be expressed in terms of exponents (x^(1/2)). This transformation allows the application of the power rule to find its derivative effectively.
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