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Chapter 2: Problem 38

Find the derivative of the function. $$ f(x)=\frac{2}{\sqrt[3]{x}}+5 \cos x $$

Short Answer

Expert verified
The derivative of the function \(f(x)=\frac{2}{\sqrt[3]{x}}+5 \cos x\) is \(f'(x) = -2/3x^{-4/3} - 5\sin(x)\).

Step by step solution

01

Rewrite the function

To prepare for differentiation, rewrite the function in a form that will make the differentiation process simpler. Rewriting the function using the power rule leads to \(f(x) = 2x^{-1/3} + 5\cos(x)\). This form allows easier application of our rules of differentiation.
02

Differentiate the first term

Apply the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Then, the derivative of \(2x^{-1/3}\) is \(-2/3 x^{-1/3-1} = -2/3x^{-4/3}\).
03

Differentiate the second term

The derivative of \(\cos(x)\) is \(-\sin(x)\). Hence, the derivative of \(5\cos(x)\) is \(5(-\sin(x)) = -5\sin(x)\).
04

Combine the results

Combine the results obtained from step 2 and step 3 to write the derivative of the total function as \(f'(x) = -2/3x^{-4/3} - 5\sin(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Rule
The power rule is a simple yet powerful tool in calculus for finding derivatives when dealing with functions of the form \(x^n\). It's particularly useful because it easily applies to both polynomial terms and power terms with fractional or negative exponents.

When using the power rule, remember:
  • The derivative of \(x^n\) is \(nx^{n-1}\).
  • Simply multiply the exponent \(n\) by the coefficient and then subtract one from the exponent \(n\).
  • This technique works regardless of whether \(n\) is positive, negative, or a fraction.

In the given exercise, we applied the power rule to the term \(2x^{-1/3}\). By following the rule:- Multiply the exponent \(-1/3\) by the coefficient 2, resulting in \(-2/3\).- Then decrease the exponent by 1, turning \(-1/3\) into \(-4/3\).

Therefore, the derivative was found to be \(-\frac{2}{3}x^{-4/3}\). Power rules simplify these actions into a straightforward process.
Trigonometric Functions
Trigonometric functions like sine and cosine occur frequently in calculus, especially when dealing with periodic or oscillating functions.

For differentiation:
  • The derivative of \(\sin(x)\) is \(\cos(x)\).
  • The derivative of \(\cos(x)\) is \(-\sin(x)\).

These derivatives showcase how trigonometric functions differ from polynomial functions after differentiation. It's important to remember the sign change that occurs with \(\cos(x)\)'s derivative.

In solving the given problem, we notices:- The term involved was \(5\cos(x)\).- Differentiating \(\cos(x)\) gives \(-\sin(x)\), and thus \(5\cos(x)\) becomes \(-5\sin(x)\) after applying constants rule of differentiation.

Mastering these simple derivatives helps in solving more complex expressions involving trigonometric functions.
Derivatives
Derivatives give us an understanding of how functions change. They act as the mathematical tool for getting the 'rate of change.'

The main concepts to grasp about derivatives include:
  • They measure how a function's value changes as its input changes.
  • Represented as \(f'(x)\) or \(\frac{df}{dx}\) for a function \(f(x)\).
  • Help in understanding the slope of the tangent line at any point on a curve.

In the problem at hand, the derivative of the function \(f(x)=\frac{2}{\sqrt[3]{x}}+5 \cos x\) was calculated using both the power rule and trigonometric differentiation.

The final derivative expression, \(f'(x) = -\frac{2}{3}x^{-4/3} - 5\sin(x)\), combines both techniques, showing how derivatives involve various forms of functions and exponents. Mastery of differentiation involves understanding how to apply such rules appropriately and effectively.

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Most popular questions from this chapter

Find an equation of the tangent line to the graph of \(g(x)=\arctan x\) when \(x=1\)

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are \(P_{1}(x)=f^{\prime}(a)(x-a)+f(a)\) and \(P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)\) \(\begin{array}{llll}\text { In Exercises } & 133-136, & \text { (a) find the specified linear and }\end{array}\) quadratic approximations of \(f,\) (b) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). \(f(x)=\tan \frac{\pi x}{4}\) \(a=1\)

Find the average rate of change of the function over the given interval. Compare this average rate of change with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=\cos x, \quad\left[0, \frac{\pi}{3}\right] $$

Find equations of both tangent lines to the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) that passes through the point (4,0).

In Exercises 37 and 38 , the derivative of the function has the same sign for all \(x\) in its domain, but the function is not one-to-one. Explain. $$ f(x)=\frac{x}{x^{2}-4} $$
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