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Chapter 3: Problem 39

Find the derivative of the following functions. $$g(t)=3 t^{2}+\frac{6}{t^{7}}$$

Short Answer

Expert verified
Answer: The derivative of the function is $$g'(t) = 6t - 42t^{-8}$$.

Step by step solution

01

Apply the sum rule

The given function is $$g(t) = 3t^2 + \frac{6}{t^7}$$. The derivative, $$g'(t)$$, is the sum of the derivatives of its components: $$g'(t) = (3t^2)' + \left(\frac{6}{t^7}\right)'$$.
02

Calculate the derivative of $$3t^2$$

We apply the power rule to find the derivative of $$3t^2$$. The derivative of $$t^2$$ is $$2t^{2-1} = 2t$$, so the derivative of $$3t^2$$ is $$3(2t) = 6t$$.
03

Calculate the derivative of $$\frac{6}{t^7}$$

First, rewrite the fraction with a negative exponent: $$\frac{6}{t^7} = 6t^{-7}$$. Now apply the power rule to find the derivative. The derivative of $$t^{-7}$$ is $$-7t^{-7-1} = -7t^{-8}$$. So, the derivative of $$6t^{-7}$$ is $$6(-7t^{-8}) = -42t^{-8}$$.
04

Combine the derivatives

The final step is to combine the derivatives found in Steps 2 and 3: $$g'(t) = 6t + (-42t^{-8}) = 6t - 42t^{-8}$$.

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Most popular questions from this chapter

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The derivative \(\frac{d}{d x}\left(e^{5}\right)\) equals \(5 \cdot e^{4}\) b. The Quotient Rule must be used to evaluate \(\frac{d}{d x}\left(\frac{x^{2}+3 x+2}{x}\right)\) c. \(\frac{d}{d x}\left(\frac{1}{x^{5}}\right)=\frac{1}{5 x^{4}}\) d. \(\frac{d^{n}}{d x^{n}}\left(e^{3 x}\right)=3^{n} \cdot e^{3 x},\) for any integer \(n \geq 1\)

Calculate the following derivatives using the Product Rule. $$\begin{array}{lll} \text { a. } \frac{d}{d x}\left(\sin ^{2} x\right) & \text { b. } \frac{d}{d x}\left(\sin ^{3} x\right) & \text { c. } \frac{d}{d x}\left(\sin ^{4} x\right) \end{array}$$ d. Based upon your answers to parts (a)-(c), make a conjecture about \(\frac{d}{d x}\left(\sin ^{n} x\right),\) where \(n\) is a positive integer. Then prove the result by induction.

Product Rule for three functions Assume that \(f, g,\) and \(h\) are differentiable at \(x\) a. Use the Product Rule (twice) to find a formula for \(\frac{d}{d x}(f(x) g(x) h(x))\) b. Use the formula in (a) to find \(\frac{d}{d x}\left(e^{2 x}(x-1)(x+3)\right)\)

An airliner passes over an airport at noon traveling \(500 \mathrm{mi} / \mathrm{hr}\) due west. At \(1: 00 \mathrm{p} . \mathrm{m} .,\) another airliner passes over the same airport at the same elevation traveling due north at \(550 \mathrm{mi} / \mathrm{hr} .\) Assuming both airliners maintain their (equal) elevations, how fast is the distance between them changing at 2: 30 p.m.?

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=3 x-4$$
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