Differentiate the following functions with respect to x: (i) sin 4x (ii) cos x3 (iii) x3 + tan x
The Correct Answer and Explanation is :
Here are the derivatives of the functions with respect to ( x ):
- For ( f(x) = \sin(4x) ):
The derivative of ( \sin(4x) ) is calculated using the chain rule, where the derivative of ( \sin(u) ) with respect to ( u ) is ( \cos(u) ), and the derivative of ( 4x ) with respect to ( x ) is 4. Hence, the derivative is:
[
\frac{d}{dx} \sin(4x) = 4 \cos(4x)
]
- For ( f(x) = \cos(x^3) ):
Again, applying the chain rule, the derivative of ( \cos(u) ) with respect to ( u ) is ( -\sin(u) ), and the derivative of ( x^3 ) with respect to ( x ) is ( 3x^2 ). Therefore, the derivative is:
[
\frac{d}{dx} \cos(x^3) = -3x^2 \sin(x^3)
]
- For ( f(x) = x^3 + \tan(x) ):
For this function, we differentiate term by term. The derivative of ( x^3 ) is ( 3x^2 ), and the derivative of ( \tan(x) ) is ( \sec^2(x) ). Hence, the derivative is:
[
\frac{d}{dx} \left(x^3 + \tan(x)\right) = 3x^2 + \sec^2(x)
]
Thus, the results for the derivatives are:
- ( 4 \cos(4x) )
- ( -3x^2 \sin(x^3) )
- ( 3x^2 + \sec^2(x) )
Let me generate an image of these derivatives for visual clarity.
Here is the image displaying the derivatives of the three functions. Let me know if you’d like further explanations or have any other questions!
