Question

Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants. $$h(t)=\frac{3}{t}+\frac{4}{t^{2}}$$

Name: Problem 36, Chapter 3: Calculus Single Variable
Uploaded: 2021-04-15T19:08:14-08:00
Duration: 1 min 17 s
Channel: Gregory Higby
Description: Find the derivatives of the given functions. Assume that a, b, c, and k are constants. h(t)=(3)/(t)+(4)/(t^2)

   Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants.
$$h(t)=\frac{3}{t}+\frac{4}{t^{2}}$$

Calculus Single Variable

Deborah… 7th Edition

Chapter 3, Problem 36

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Solved by Expert Gregory Higby

04/15/2021

Step 1

This will make it easier to apply the power rule for derivatives. The function $h(t)$ can be rewritten as: $$h(t)=3t^{-1}+4t^{-2}$$ Show more…

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Find the derivatives of the given functions. Assume that $a, b, c,$ and $k$ are constants. $$h(t)=\frac{3}{t}+\frac{4}{t^{2}}$$

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Key Concepts

Rewriting Rational Functions as Power Functions

Rational functions, such as those in the form 3/t or 4/t^2, can be rewritten using negative exponents (e.g., 3*t^(-1) and 4*t^(-2)). This transformation allows the application of the power rule directly, making the differentiation process more straightforward.

Power Rule

The power rule is a fundamental principle in calculus that is used to differentiate functions of the form f(t) = t^n. According to this rule, the derivative of t^n is n*t^(n-1). This rule is especially useful when functions are expressed with exponents, including negative exponents after rewriting rational functions.

Differentiation of Constant Multiples

When differentiating a function that is multiplied by a constant, the constant factor remains unchanged and can be pulled out of the derivative. This property is important because it simplifies the differentiation process by allowing the focus to be on the variable part of the function.

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