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Chapter 3: Problem 19

Find \(d y / d x\) for the following functions. $$y=5 x^{2}+\cos x$$

Short Answer

Expert verified
Answer: The derivative of the function is \(\frac{dy}{dx} = 10x - \sin x\).

Step by step solution

01

Identify the individual terms in the function

We'll start by identifying the two terms in the given function. The function is represented as a sum of two terms: 1. \(5x^2\) 2. \(\cos x\)
02

Apply the power rule for differentiation

Next, we'll apply the power rule for differentiation to the first term, which is \(5x^2\). The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). For our term \(5x^2\), we have \(n=2\). So, we get: $$\frac{d}{dx}(5x^2) = 2 \cdot 5x^{2-1} = 10x^1 = 10x$$
03

Differentiate the cosine function

Now, we'll find the derivative of the second term, which is \(\cos x\). The derivative of the cosine function is given by: $$\frac{d}{dx}(\cos x) = -\sin x$$
04

Combine the results of the derivatives

We found the derivatives of the individual terms in the function. Now, we need to combine the results to find the derivative of the complete function. Since the original function is the sum of the two terms, we simply add the derivatives of those terms to get the final derivative: $$\frac{dy}{dx} = 10x - \sin x$$ Thus, the derivative of the function \(y = 5x^2 + \cos x\) with respect to x is: $$\frac{dy}{dx} = 10x - \sin x$$

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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 in calculus is a straightforward tool for differentiating functions of the form \(x^n\). It simplifies the process by providing a quick formula. Here is how it works:
  • When you have a term \(x^n\), the power rule tells us to multiply the term by the exponent \(n\).
  • Then, you reduce the exponent by 1.
For example, if you have \(5x^2\), you will multiply 5 by the exponent 2, giving you 10.
Then, you reduce the power of \(x\) from 2 to 1. So, the derivative of \(5x^2\) becomes \(10x\).
This rule is very useful for differentiating polynomial expressions, which are made up of terms like \(x^n\).
It also applies whether the exponent is negative, positive, or even a fraction!
Understanding the power rule can greatly speed up the process of finding derivatives.
Trigonometric Derivatives
Understanding the derivative of trigonometric functions is crucial for solving calculus problems involving these functions. They have specific rules just like polynomial terms.
  • For cosine, the derivative is \(-\sin x\).
  • For sine, the derivative is \(\cos x\).
These rules might seem like memorization, but they arise from the particular shapes and angles of the unit circle.
When differentiating \(\cos x\), you apply the rule for cosine directly, yielding \(-\sin x\).
This means that wherever you see \(\cos x\), differentiate it to \(-\sin x\) when solving problems.
Remembering these two basic trigonometric derivatives will help you tackle more complex problems involving trigonometric functions.
Derivative of Polynomial Functions
Polynomial functions are perhaps the easiest to differentiate because they follow the power rule so well.
A polynomial is a sum of terms with the form \(ax^n\).
  • Each term can be differentiated individually by applying the power rule.
  • The derivative of a constant term (with \(n=0\)) is always 0.
In the exercise given, the function \(y = 5x^2 + \cos x\) is a sum of a polynomial part \(5x^2\) and a trigonometric part \(\cos x\).
We differentiate each part separately. The polynomial \(5x^2\) becomes \(10x\) using the power rule.
The trigonometric term becomes \(-\sin x\).
By understanding how each piece works, you can approach any polynomial鈥攐r combination of polynomial and other functions鈥攚ith confidence. Begin by isolating each term, apply the rules, and sum the results.
This step-by-step method makes handling polynomials in calculus much simpler.

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Most popular questions from this chapter

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$y^{2}-3 x y=2$$

The bottom of a large theater screen is \(3 \mathrm{ft}\) above your eye level and the top of the screen is \(10 \mathrm{ft}\) above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of \(3 \mathrm{ft} / \mathrm{s}\) while looking at the screen. What is the rate of change of the viewing angle \(\theta\) when you are \(30 \mathrm{ft}\) from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Vertical tangent lines a. Determine the points where the curve \(x+y^{2}-y=1\) has a vertical tangent line (see Exercise 53 ). b. Does the curve have any horizontal tangent lines? Explain.

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(x^{10 x}\right)$$

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x^{2}\left(3 y^{2}-2 y^{3}\right)=4$$
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