Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
Find the derivative -do not simplify-please look at picture-thanks
![**Mathematics: Calculus - Derivatives**
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**Problem Statement:**
Find the derivative of each function. Do not simplify.
1. \( f(x) = (x^3 + \sqrt{\cos x})(\sec x) \)
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*Explanation:*
To find the derivative of the given function \( f(x) = (x^3 + \sqrt{\cos x})(\sec x) \), one would typically use the product rule from calculus. The product rule states that if you have a function that is the product of two functions, say \(u(x)\) and \(v(x)\), then the derivative \(f(x)\) is given by:
\[f'(x) = u'(x)v(x) + u(x)v'(x)\]
Here, let:
\( u(x) = x^3 + \sqrt{\cos x} \)
and
\( v(x) = \sec x \)
Calculate the derivatives \( u'(x) \) and \( v'(x) \), and then apply the product rule to find \( f'(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb1340680-eb9c-48d5-bc71-d4a4fad49dd5%2F6041bbaf-52a2-46f4-944f-c7daef15e748%2F5ceg9gb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematics: Calculus - Derivatives**
---
**Problem Statement:**
Find the derivative of each function. Do not simplify.
1. \( f(x) = (x^3 + \sqrt{\cos x})(\sec x) \)
---
*Explanation:*
To find the derivative of the given function \( f(x) = (x^3 + \sqrt{\cos x})(\sec x) \), one would typically use the product rule from calculus. The product rule states that if you have a function that is the product of two functions, say \(u(x)\) and \(v(x)\), then the derivative \(f(x)\) is given by:
\[f'(x) = u'(x)v(x) + u(x)v'(x)\]
Here, let:
\( u(x) = x^3 + \sqrt{\cos x} \)
and
\( v(x) = \sec x \)
Calculate the derivatives \( u'(x) \) and \( v'(x) \), and then apply the product rule to find \( f'(x) \).
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