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

Identify Functions: We need to find the derivative of the function g(x) = e^(tan(x)). To do this, we will use the chain rule, which 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.Derivative of Outer Function: First, let's identify the outer and inner functions. The outer function is e^u, where u is the inner function. The inner function is tan(x).Derivative of Inner Function: Now, we take the derivative of the outer function with respect to the inner function. The derivative of e^u with respect to u is e^u. (d/du)(e^u) = e^uApply Chain Rule: Next, we take the derivative of the inner function tan(x) with respect to x. The derivative of tan(x) is sec^2(x). (d/dx)(tan(x)) = sec^2(x)Match with Options: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. g'(x) = (d/du)(e^u) * (d/dx)(tan(x)) g'(x) = e^(tan(x)) * sec^2(x)We can now match our result with the given options. The correct answer is: g'(x) = e^(tan(x)) * sec^2(x) This corresponds to option (C).

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

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

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

\newline

(B)

e^{\tan (x)}

\newline

(C)

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

\newline

(D)

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

Identify Functions: We need to find the derivative of the function $g(x) = e^{\tan(x)}$ . To do this, we will use the chain rule, which 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.
Derivative of Outer Function: First, let's identify the outer and inner functions. The outer function is $e^u$ , where $u$ is the inner function. The inner function is $\tan(x)$ .
Derivative of Inner Function: Now, we take the derivative of the outer function with respect to the inner function. The derivative of $e^u$ with respect to $u$ is $e^u$ . $\newline$ $\frac{d}{du}(e^u) = e^u$
Apply Chain Rule: Next, we take the derivative of the inner function $\tan(x)$ with respect to $x$ . The derivative of $\tan(x)$ is $\sec^2(x)$ . $\newline$ $(\frac{d}{dx})(\tan(x)) = \sec^2(x)$
Match with Options: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. $\newline$ $g'(x) = \frac{d}{du}(e^u) \cdot \frac{d}{dx}(\tan(x))$ $\newline$ $g'(x) = e^{\tan(x)} \cdot \sec^2(x)$
Match with Options: Applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function. $\newline$ $g'(x) = \frac{d}{du}(e^u) \cdot \frac{d}{dx}(\tan(x))$ $\newline$ $g'(x) = e^{\tan(x)} \cdot \sec^2(x)$ We can now match our result with the given options. The correct answer is: $\newline$ $g'(x) = e^{\tan(x)} \cdot \sec^2(x)$ $\newline$ This corresponds to option (C).