
Differentiate the following with respect to \[x\]: \[{e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
Answer
458.7k+ views
Hint: Here, we will find the differentiation of the given function with respect to the variable. We will find the derivative of all the functions separately. First, we will find the derivative of the exponential function and then the derivative of the exponential of the trigonometric function. Then we will simplify it further to get the required value.
Formula Used:
We will use the following formulas:
1. \[\dfrac{d}{{dx}}\left( {{e^x}} \right) = x{e^x}\]
2. \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]
3. \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\]
4. \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]
5. \[2\sin x\cos x = \sin 2x\]
Complete step-by-step answer:
Let the given function be \[y\].
So, we get \[y = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
Differentiating with respect to \[x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{e^{3{{\sin }^2}x - 2{{\cos }^2}x}}} \right)\]
By using the derivative formula \[\dfrac{d}{{dx}}\left( {{e^x}} \right) = x{e^x}\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}} \cdot \dfrac{d}{{dx}}\left( {3{{\sin }^2}x - 2{{\cos }^2}x} \right)\]
We know that \[{\sin ^2}x = {\left( {\sin x} \right)^2}\].
So, rewriting the above equation, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}} \cdot \left[ {3\dfrac{d}{{dx}}{{\left( {\sin x} \right)}^2} - 2\dfrac{d}{{dx}}{{\left( {\cos x} \right)}^2}} \right]\]
By using the derivative formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}} \cdot \left[ {3 \cdot 2\sin x \cdot \dfrac{d}{{dx}}\left( {\sin x} \right) - 2 \cdot 2\cos x\dfrac{d}{{dx}}\left( {\cos x} \right)} \right]\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {6\sin x \cdot \dfrac{d}{{dx}}\left( {\sin x} \right) - 4\cos x\dfrac{d}{{dx}}\left( {\cos x} \right)} \right]\]
By using the derivative formula \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\] and \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\] , we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {6\sin x \cdot \cos x - 4\cos x \cdot \left( { - \sin x} \right)} \right]\]
We know that the product of two negative integers is a positive integer. Therefore, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {6\sin x \cdot \cos x + 4\sin x \cdot \cos x} \right]\]
By adding the terms, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {10\sin x \cdot \cos x} \right]\]
By rewriting the terms in terms of the trigonometric identity, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {5 \cdot 2\sin x\cos x} \right]\]
Using the trigonometric formula, \[2\sin x\cos x = \sin 2x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {5\sin 2x} \right]\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 5\sin 2x \cdot {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
\[ \Rightarrow y' = 5\sin 2x \cdot {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
Therefore, the derivative of \[{e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\] is \[5\sin 2x \cdot {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\].
Note: We should know that if a function has two functions, then both the functions have to be differentiated separately. Differentiation is a method of finding the derivative the function and finding the rate of change of function with respect to one. Here we have found out the derivative of exponential function. Exponential function is a constant which is raised to some power. Exponential function is the inverse of logarithmic function.
Formula Used:
We will use the following formulas:
1. \[\dfrac{d}{{dx}}\left( {{e^x}} \right) = x{e^x}\]
2. \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]
3. \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\]
4. \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]
5. \[2\sin x\cos x = \sin 2x\]
Complete step-by-step answer:
Let the given function be \[y\].
So, we get \[y = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
Differentiating with respect to \[x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {{e^{3{{\sin }^2}x - 2{{\cos }^2}x}}} \right)\]
By using the derivative formula \[\dfrac{d}{{dx}}\left( {{e^x}} \right) = x{e^x}\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}} \cdot \dfrac{d}{{dx}}\left( {3{{\sin }^2}x - 2{{\cos }^2}x} \right)\]
We know that \[{\sin ^2}x = {\left( {\sin x} \right)^2}\].
So, rewriting the above equation, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}} \cdot \left[ {3\dfrac{d}{{dx}}{{\left( {\sin x} \right)}^2} - 2\dfrac{d}{{dx}}{{\left( {\cos x} \right)}^2}} \right]\]
By using the derivative formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}} \cdot \left[ {3 \cdot 2\sin x \cdot \dfrac{d}{{dx}}\left( {\sin x} \right) - 2 \cdot 2\cos x\dfrac{d}{{dx}}\left( {\cos x} \right)} \right]\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {6\sin x \cdot \dfrac{d}{{dx}}\left( {\sin x} \right) - 4\cos x\dfrac{d}{{dx}}\left( {\cos x} \right)} \right]\]
By using the derivative formula \[\dfrac{d}{{dx}}\left( {\sin x} \right) = \cos x\] and \[\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x\] , we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {6\sin x \cdot \cos x - 4\cos x \cdot \left( { - \sin x} \right)} \right]\]
We know that the product of two negative integers is a positive integer. Therefore, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {6\sin x \cdot \cos x + 4\sin x \cdot \cos x} \right]\]
By adding the terms, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {10\sin x \cdot \cos x} \right]\]
By rewriting the terms in terms of the trigonometric identity, we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {5 \cdot 2\sin x\cos x} \right]\]
Using the trigonometric formula, \[2\sin x\cos x = \sin 2x\], we get
\[ \Rightarrow \dfrac{{dy}}{{dx}} = {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\left[ {5\sin 2x} \right]\]
\[ \Rightarrow \dfrac{{dy}}{{dx}} = 5\sin 2x \cdot {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
\[ \Rightarrow y' = 5\sin 2x \cdot {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\]
Therefore, the derivative of \[{e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\] is \[5\sin 2x \cdot {e^{3{{\sin }^2}x - 2{{\cos }^2}x}}\].
Note: We should know that if a function has two functions, then both the functions have to be differentiated separately. Differentiation is a method of finding the derivative the function and finding the rate of change of function with respect to one. Here we have found out the derivative of exponential function. Exponential function is a constant which is raised to some power. Exponential function is the inverse of logarithmic function.
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Trending doubts
10 examples of friction in our daily life

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Prokaryotic Cells and Eukaryotic Cells

State and prove Bernoullis theorem class 11 physics CBSE

What organs are located on the left side of your body class 11 biology CBSE

Write down 5 differences between Ntype and Ptype s class 11 physics CBSE
