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Chapter 16: Problem 25

Find the derivatives of the given functions. \(v(x)=\sec \left(x^{2.2}+1.2 x-1\right)\)

Short Answer

Expert verified
The derivative of the given function is: \(v'(x) = (\sec(x^{2.2} + 1.2x - 1)\tan(x^{2.2} + 1.2x - 1)) \cdot (2.2x^{1.2} + 1.2)\)

Step by step solution

01

Differentiate the inner function u(x)

Find the derivative of \(u(x) = x^{2.2} + 1.2x - 1\) with respect to x. We can differentiate each term separately: \(\frac{d}{dx}(x^{2.2}) = 2.2x^{1.2}\) \(\frac{d}{dx}(1.2x) = 1.2\) \(\frac{d}{dx}(-1) = 0\) Now add these derivatives together: \(\frac{du}{dx} = 2.2x^{1.2} + 1.2\)
02

Differentiate the outer function v(x)

Now we need to differentiate the outer function \(v(x) = \sec(u(x))\). We know that the derivative of the secant function is: \(\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)\) So, the derivative of \(v(x)\) with respect to \(u(x)\) is: \(\frac{dv}{du} = \sec(u(x))\tan(u(x))\)
03

Use the Chain Rule to find the derivative of v(x)

Now we can use the chain rule to combine the derivatives from steps 1 and 2: \(\frac{dv}{dx} = \frac{dv}{du} \cdot \frac{du}{dx}\) Substitute the derivatives from steps 1 and 2: \(v'(x) = (\sec(u(x))\tan(u(x))) \cdot (2.2x^{1.2} + 1.2)\) Finally, substitute the original expression for \(u(x)\) back in and simplify:
04

Simplify the derivative

\(v'(x) = (\sec(x^{2.2} + 1.2x - 1)\tan(x^{2.2} + 1.2x - 1)) \cdot (2.2x^{1.2} + 1.2)\) The derivative of the given function is: \(v'(x) = (\sec(x^{2.2} + 1.2x - 1)\tan(x^{2.2} + 1.2x - 1)) \cdot (2.2x^{1.2} + 1.2)\)

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Key Concepts

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

Chain Rule
The chain rule is an essential concept in calculus for finding the derivative of composite functions. A composite function is when one function is nested inside another, like in our problem where we have the \(\sec(u(x))\)function with \(u(x)\) as \(x^{2.2} + 1.2x - 1\). Here's how it works:
  • First, find the derivative of the inner function, which is \(u(x)\) in this case.
  • Then, find the derivative of the outer function as if the inner function was just a simple variable.
  • Lastly, multiply these derivatives together to get the derivative of the entire composite function.
By following these steps, you apply the chain rule to successfully find the derivative of composite functions.
Secant Function
The secant function, denoted as \(\sec(x)\), is one of the trigonometric functions and is defined as the reciprocal of the cosine function \(\frac{1}{\cos(x)}\). Understanding its derivative is crucial in calculus problems involving trigonometric expressions.
  • The derivative of the secant function \(\sec(x)\) with respect to \(x\) is \(\sec(x)\tan(x)\).
  • This tells us how the secant function changes as \(x\) changes.
In our problem, after finding the derivative of the inner function \(u(x)\), we used the derivative of the secant function, allowing us to apply the chain rule accurately for the overall derivative.
Calculus
Calculus is a branch of mathematics that studies how quantities change. It is fundamental for understanding rates of change and has two main branches: differential calculus and integral calculus.
Our focus here is on differential calculus, specifically finding derivatives. Derivatives measure how a function's value changes concerning one of its input variables.
  • The process of finding a derivative is called differentiation.
  • We often utilize tools like the chain rule, product rule, and quotient rule to differentiate more complex functions.
Through the exercise, we used calculus to determine how the given function \(v(x) = \sec(x^{2.2} + 1.2x - 1)\)changes with respect to \(x\).In essence, calculus allows us to dissect complex expressions into manageable calculations of small-scale changes, paving the way for deeper mathematical insights.

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