Answer

Verified

428.1k+ views

**Hint**: First, before proceeding for this, we must know the following formula to calculate the derivative of the exponential function. Then, by using the same concept of the chain rule to the given function, we get the desired answer.

**:**

__Complete step-by-step answer__In this question, we are supposed to find the slope of the curve whose equation is given as $ y=m{{e}^{\sin x}} $ shown in the graph where P is the point which cuts the y-axis.

So, before proceeding for this, we must know the following formula to calculate the derivative of the exponential function.

Now, let us consider any exponential function as $ {{e}^{ax}} $ .

Then, to get the derivative of the above exponential function we have:

$ \dfrac{d}{dx}\left( {{e}^{ax}} \right)=a{{e}^{ax}} $

Now, by using the same concept of the chain rule to the given function.

Also, we will get to know another factor that the slope of the curve is always the derivative of the given function.

So, the slope of the given function as $ y=m{{e}^{\sin x}} $ where m>0 is :

$ \dfrac{dy}{dx}=\dfrac{d}{dx}\left( m{{e}^{\sin x}} \right) $

Then, by solving for the value of slope of the function by using the chain rule as:

$ \begin{align}

& \dfrac{dy}{dx}=m{{e}^{\sin x}}\times \dfrac{d}{dx}\left( \sin x \right) \\

& \dfrac{dy}{dx}=m\cos x{{e}^{\sin x}} \\

\end{align} $

So, by calculation we get the slope as $ m\cos x{{e}^{\sin x}} $ .

Hence, the slope of the function $ y=m{{e}^{\sin x}} $ is $ m\cos x{{e}^{\sin x}} $ .

**Note**: Now, to solve these types of questions we need to know some of the basic derivatives of the function to get the answer appropriately. So, some of the most important derivatives of the function are as:

$ \begin{align}

& \dfrac{d}{dx}\left( \sin x \right)=\cos x \\

& \dfrac{d}{dx}\left( \cos x \right)=-\sin x \\

& \dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}} \\

\end{align} $

Moreover, we must be careful while applying the chain rule so that we will not miss any term till the end. As in the above question we may write $ m{{e}^{\sin x}} $ as the final slope and forget to take the derivative of the function $ \sin x $ .

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE