Given f(x)=2sec(2x), find f^(')(x). Answer: f^(')(x)=

Apply Chain Rule: To find the derivative of the function f(x) = 2sec(2x), we need to apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Identify Outer and Inner Functions: The outer function is sec(u) where u = 2x, and the inner function is 2x. The derivative of sec(u) with respect to u is sec(u)tan(u), and the derivative of 2x with respect to x is 2.Use Chain Rule Formula: Now we apply the chain rule: f'(x) = d/dx[2sec(2x)] = 2 * d/dx[sec(2x)] = 2 * sec(2x)tan(2x) * d/dx[2x].Substitute Inner Function Derivative: We already determined that d/dx[2x] is 2, so we can substitute this into our expression: f'(x) = 2 * sec(2x)tan(2x) * 2.Simplify Expression: Simplify the expression by multiplying the constants: f'(x) = 4sec(2x)tan(2x).

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Given
f(x)=2sec(2x), find
f^(')(x).
Answer:
f^(')(x)=

Given $f(x)=2 \sec (2 x)$ , find $f^{\prime}(x)$ . $\newline$ Answer: $f^{\prime}(x)=$

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Math Problems

Grade 8

Evaluate a nonlinear function

Full solution

Q. Given

f(x)=2 \sec (2 x)

, find

f^{\prime}(x)

\newline

Answer:

f^{\prime}(x)=

Apply Chain Rule: To find the derivative of the function $f(x) = 2\sec(2x)$ , we need to apply the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Identify Outer and Inner Functions: The outer function is $\sec(u)$ where $u = 2x$ , and the inner function is $2x$ . The derivative of $\sec(u)$ with respect to $u$ is $\sec(u)\tan(u)$ , and the derivative of $2x$ with respect to $x$ is $2$ .
Use Chain Rule Formula: Now we apply the chain rule: $f'(x) = \frac{d}{dx}[2\sec(2x)] = 2 \cdot \frac{d}{dx}[\sec(2x)] = 2 \cdot \sec(2x)\tan(2x) \cdot \frac{d}{dx}[2x].$
Substitute Inner Function Derivative: We already determined that $\frac{d}{dx}[2x]$ is $2$ , so we can substitute this into our expression: $f'(x) = 2 \cdot \sec(2x)\tan(2x) \cdot 2$ .
Simplify Expression: Simplify the expression by multiplying the constants: $f'(x) = 4\sec(2x)\tan(2x)$ .