Let g(x)=e^(tan(x)). Find g^(')(x). Choose 1 answer: (A) e^(tan(x)) (B) e^(tan(x))sec^(2)(x) (C) tan(x)e^(tan(x)-1) (D) e^(sec^(2)(x))

Apply Chain Rule: Apply the chain rule to differentiate g(x) = e^(tan(x)). 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 Functions: Identify the outer and inner functions. The outer function is e^u, where u is the inner function. The inner function is tan(x).Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of e^u with respect to u is e^u. So, (d/du)(e^u) = e^u.Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of tan(x) with respect to x is sec^2(x). So, (d/dx)(tan(x)) = sec^2(x).Apply Chain Rule Again: Apply the chain rule using the derivatives from steps 3 and 4. g'(x) = (d/du)(e^u) * (d/dx)(tan(x)) g'(x) = e^(tan(x)) * sec^2(x).Match Answer Choices: Match the result with the given answer choices. The correct answer is (B) e^(tan(x))sec^(2)(x).

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Let
g(x)=e^(tan(x)).
Find
g^(')(x).
Choose 1 answer:
(A)
e^(tan(x))
(B)
e^(tan(x))sec^(2)(x)
(C)
tan(x)e^(tan(x)-1)
(D)
e^(sec^(2)(x))

Let $g(x)=e^{\tan (x)}$ . $\newline$ Find $g^{\prime}(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $e^{\tan (x)}$ $\newline$ (B) $e^{\tan (x)} \sec ^{2}(x)$ $\newline$ (C) $\tan (x) e^{\tan (x)-1}$ $\newline$ (D) $e^{\sec ^{2}(x)}$

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Let

g(x)=e^{\tan (x)}

\newline

Find

g^{\prime}(x)

\newline

Choose

1

answer:

\newline

(A)

e^{\tan (x)}

\newline

(B)

e^{\tan (x)} \sec ^{2}(x)

\newline

(C)

\tan (x) e^{\tan (x)-1}

\newline

(D)

e^{\sec ^{2}(x)}

Apply Chain Rule: Apply the chain rule to differentiate $g(x) = e^{\tan(x)}$ . $\newline$ 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 Functions: Identify the outer and inner functions. $\newline$ The outer function is $e^u$ , where $u$ is the inner function. $\newline$ The inner function is $\tan(x)$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $e^u$ with respect to $u$ is $e^u$ . $\newline$ So, $(\frac{d}{du})(e^u) = e^u$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $\tan(x)$ with respect to $x$ is $\sec^2(x)$ . So, $(\frac{d}{dx})(\tan(x)) = \sec^2(x)$ .
Apply Chain Rule Again: Apply the chain rule using the derivatives from steps $3$ and $4$ . $\newline$ $g'(x) = \frac{d}{du}(e^u) \cdot \frac{d}{dx}(\tan(x))$ $\newline$ $g'(x) = e^{\tan(x)} \cdot \sec^2(x)$ .
Match Answer Choices: Match the result with the given answer choices. $\newline$ The correct answer is (B) $e^{\tan(x)}\sec^{2}(x)$ .