Calculate the derivative indicated. .(d^4 f)/(d t^4)|t=1, f(t)=6 t^9-2 t^5

Name: Problem 20, Chapter 3: Calculus: Early Transcendentals
Uploaded: 2021-07-10T18:23:08-08:00
Duration: 1 min 43 s
Channel: Wesley Hines
Description: Calculate the derivative indicated. .(d^4 f)/(d t^4)|t=1, f(t)=6 t^9-2 t^5

step 1 For this problem, we're going to be calculating the fourth derivative at t equals 1 of 6 t to th

Calculate the derivative indicated. .(d^4 f)/(d t^4)|t=1, ...

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Key Concepts

Polynomial Differentiation

Polynomial differentiation involves finding the derivative of a polynomial function term-by-term using basic differentiation rules, such as the power rule. This technique simplifies the process since each term in a polynomial can be separately differentiated, especially when dealing with functions of the form at^n.

Power Rule

The power rule is a fundamental differentiation technique used to differentiate functions of the form t^n, where n is a real number. According to this rule, the derivative of t^n is n*t^(n-1), and it is applied repeatedly for each term in a polynomial.

Higher-Order Derivatives

Higher-order derivatives are derivatives taken more than once, representing the rate of change of a function’s rate of change. The computation of the fourth derivative, or any nth derivative, involves applying differentiation rules iteratively, simplifying the expression at each step.

Evaluation of Derivatives at Specific Points

After computing the derivative, evaluating it at a specific point involves substituting the value into the derived expression. This is often used to understand the behavior of a function at a particular point, such as analyzing the curvature or acceleration.

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