(B) Let $y = e^{\cos x}$.Using the chain rule,we differentiate with respect to $x$:$\frac{dy}{dx} = \frac{d}{dx}(e^{\cos x})$Since $\frac{d}{du}(e^u)

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(B) Let $y = e^{\cos x}$.Using the chain rule,we differentiate with respect to $x$:$\frac{dy}{dx} = \frac{d}{dx}(e^{\cos x})$Since $\frac{d}{du}(e^u) = e^u$ and $\frac{d}{dx}(\cos x) = -\sin x$,we apply the chain rule:$\frac{dy}{dx} = e^{\cos x} \cdot \frac{d}{dx}(\cos x)$$\frac{dy}{dx} = e^{\cos x} \cdot (-\sin x)$$\frac{dy}{dx} = -(\sin x) e^{\cos x}$

Class 12 Mathematics Continuity and Differentiability Derivative at a point, Standard differentiation

Easy

Differentiate the following with respect to $x$: $e^{\cos x}$

A
$e^{\cos x} \sin x$
B
$-e^{\cos x} \sin x$
C
$e^{\cos x} \cos x$
D
$-e^{\cos x} \cos x$

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Class 12

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Continuity and Differentiability

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Derivative at a point, Standard differentiation

Differentiate the following with respect to $x$: $e^{x}+e^{x^{2}}+e^{x^{3}}+e^{x^{4}}+e^{x^{5}}$

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Compute the derivative of $6 x^{100}-x^{55}+x$.

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Differentiate the following with respect to $x$: $e^{\cos x}$

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Differentiate the following with respect to $x$: $e^{x}+e^{x^{2}}+e^{x^{3}}+e^{x^{4}}+e^{x^{5}}$

The left-hand derivative of $f(x) = \{x\} \sin(\pi x)$ at $x = k$ ($k$ is an integer) is (where $\{ \}$ denotes the fractional part function).

Let $f(x + y) = f(x)f(y)$ and $f(x) = 1 + \sin(3x)g(x)$,where $g(x)$ is continuous. Then $f'(x)$ is:

If $\frac{d}{d x}\left\{\left(\frac{x-1}{x-\sqrt{x}}\right) e^{2 x+1}\right\}=\frac{x-1}{x-\sqrt{x}} e^{2 x+1} f(x)$, then $f(4)=$

Compute the derivative of $6 x^{100}-x^{55}+x$.

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