# Derivative of e^cos x

The derivative of ecos x is -sin x ecos x. In this post, we will calculate the derivative of e to the power cos x by the first principle of derivatives as well as by the logarithmic differentiation and the chain rule of derivatives. We will also solve a few questions related to the differentiation of ecos x.

## What is the derivative of ecos x?

To find the derivative of e to the power cos x, we will first apply the chain rule of derivatives. The chain rule says that if f is a function of u, then the derivative of f with respect to x is

$\dfrac{df}{dx}=\dfrac{df}{du} \cdot \dfrac{du}{dx}$.

Step 1: Note that ecos x is a function of cos x. So here f(u) = eu with u=cos x.

Step 2: As u = cos x, we have

$\dfrac{du}{dx}=-\sin x$

Step 3: Now, by the above chain rule of derivatives, the derivative of ecos x is

$\dfrac{d}{dx}\left(e^{\cos x} \right)=\dfrac{d}{dx}\left(e^u \right)$

$=\dfrac{d}{du}\left(e^u \right) \cdot \dfrac{du}{dx}$

$=e^u \cdot (-\sin x)$ (putting the value of du/dx from above)

$=-\sin x e^{\cos x}$ as u=cos x.

So the derivative of ecos x is -sin x esin x. In other words, d/dx(ecos x)=-sin x e cos x.

## Derivative of ecos x from first principle

Now we will find the derivative of ecos x using the first principle of derivatives. The derivative of a function f(x) by first principle is given by the following limit:

$\dfrac{d}{dx}\left( f(x) \right)$ $= \lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$

Let f(x)=ecos x. So the derivative of ecos x by first principle is

$\dfrac{d}{dx}\left( e^{\cos x} \right)$ $= \lim\limits_{h \to 0} \dfrac{e^{\cos(x+h)} – e^{\cos x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{\cos x}(e^{\cos(x+h)-\cos x} -1)}{h}$

$=e^{\cos x}\lim\limits_{h \to 0} \dfrac{e^{\cos(x+h)-\cos x} -1}{h}$

$=e^{\cos x}\lim\limits_{h \to 0} \dfrac{e^{\cos(x+h)-\cos x} -1}{\cos(x+h)-\cos x}$ $\times \lim\limits_{h \to 0} \dfrac{\cos(x+h)-\cos x}{h}$

[Let t=cos(x+h)-cos x. Then t→0 as h→0]

$=e^{\cos x} \lim\limits_{t \to 0} \dfrac{e^t-1}{t}$ $\times \lim\limits_{h \to 0} \dfrac{-2\sin(x+h/2)\sin h/2}{h}$ as we know that cos c – cos d= -2 sin(c+d)/2 sin(c-d)/2.

$=-e^{\cos x} \times 1$ $\times \lim\limits_{h \to 0}\sin(x+h/2)$ $\times \lim\limits_{h \to 0} \dfrac{\sin h/2}{h/2}$ as the limit of (et-1)/t is 1 when t tends to 0.

$=-e^{\cos x} \sin(x+0/2)$ $\times \lim\limits_{z \to 0} \dfrac{\sin z}{z}$ where z=h/2.

$=-e^{\cos x} \sin x \times 1$ as $\lim\limits_{x \to 0} \dfrac{\sin x}{x}=1$

$=-\sin x e^{\cos x}$

Hence the derivative of ecos x is equal to -sin x ecos x , obtained by the first principle of derivatives.

## Derivative of ecos x by substitution

Next, we evaluate the derivative of e to the power cos x by the substitution method. The logarithmic derivatives will be used here.

Step 1: Let u=ecos x. We need to find du/dx.

Step 2: Taking logarithm on both sides, we get

logeu = logeecos x

⇒ logeu = cos x logee

⇒ logeu = cos x as we know that logaa=1.

Step 3: Differentiating with respect to x, we get that

$\dfrac{d}{dx}\left(\log_e u \right)=\dfrac{d}{dx}\left(\cos x \right)$

⇒ $\dfrac{1}{u} \dfrac{du}{dx}=-\sin x$

⇒ $\dfrac{du}{dx}=-u \sin x =-e^{\cos x} \sin x$ as u=ecos x

Therefore, we obtain the derivative of e power cos x which is -sin x ecos x.

## Question Answer on Derivative of ecos x

Question 1: What is the derivative of xecos x?

By product rule, the derivative of xecos x is

= x d/dx(ecos x) + ecos x d/dx(x)

= x (-sin xecos x) + ecos x $\cdot$ 1

= -x sin xecos x + ecos x

= ecos x (-x sin x+1)

Question 2: Find the second derivative of ecos x?

As the first derivative of ecos x is -sin x ecos x, the second derivative of e to the power cos x is equal to

= d/dx(-sin x ecos x)

= -[sin x d/dx(ecos x)+ ecos x d/dx(sin x)] by the product rule of derivatives.

= -[sin x (-sin x ecos x)+ecos x(-cos x)]

= (sin2x-cos x)ecos x

So the second order derivative of e to the power cos x is (sin2x-cos x)ecos x.

## FAQs on Derivative of ecos x

Q1: What is the derivative of ecos x?

Answer: The derivative of ecos x is ecos x × d/dx(cos x) = -sin x ecos x. Here we have applied the chain rule of derivatives.

Q2: What is the derivative of esin x?

Answer: The derivative of esin x is esin x × d/dx(sin x) = cos x esin x.

Q3: What is the derivative of ecos (x^2)?

Answer: The derivative of ecos (x^2) is ecos (x^2) × d/dx(cos x^2) = -sin(x^2) ecos(x^2) d/dx(x^2) = -2x sin(x^2) ecos(x^2). Here we have used the derivative of cos x is -sin x and the derivative of x2 is 2x.