Answer :
To find the derivative of the function [tex]\(e^{-7x^6}\)[/tex], we will employ the chain rule of differentiation. The chain rule is used when differentiating composite functions. Therefore, the derivative of [tex]\(e^{-7x^6}\)[/tex] with respect to x is: [tex]\[\frac{d}{dx}(e^{-7x^6}) = -42x^5 \cdot e^{-7x^6}\][/tex]
To find the derivative of the function [tex]\(e^{-7x^6}\)[/tex], we will employ the chain rule of differentiation. The chain rule is used when differentiating composite functions.
The general form of the chain rule is:
[tex]\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\][/tex]
In our case, the function:
[tex]\(f(u) = e^u\)[/tex]
Where:
[tex]\(u = -7x^6\)[/tex]
[tex]\(u = -7x^6\)[/tex] is a composite function with:
[tex]\(g(x) = -7x^6\)[/tex]
Here,
[tex](f'(u) = e^u\)[/tex]
The derivative of [tex]\(e^u\)[/tex] with respect to u is:
[tex]\(e^u\)[/tex]
[tex]\(g'(x) = -7 \cdot 6x^5\)[/tex]
The derivative of [tex]\(x^n\)[/tex] with respect to x is:
[tex]\(nx^{n-1}\)[/tex]
Now let's apply the chain rule:
[tex]\[\frac{d}{dx}[e^{-7x^6}] = \frac{d}{du}[e^u] \cdot \frac{d}{dx}[-7x^6]\][/tex]
We've established that [tex]\(f'(u) = e^u\)[/tex], and we now need to find g'(x):
[tex]\[g'(x) = \frac{d}{dx}[-7x^6] = -7 \cdot 6x^{6-1} = -42x^5\][/tex]
So, using the derivatives of \(f\) and \(g\), we get:
[tex]\[\frac{d}{dx}[e^{-7x^6}] = e^u \cdot (-42x^5)\][/tex]
Don't forget to substitute back for u, which is [tex]\(-7x^6\)[/tex]:
[tex]\[\frac{d}{dx}[e^{-7x^6}] = e^{-7x^6} \cdot (-42x^5)\][/tex]
Therefore, the derivative of [tex]\(e^{-7x^6}\)[/tex] with respect to x is:
[tex]\[\frac{d}{dx}(e^{-7x^6}) = -42x^5 \cdot e^{-7x^6}\][/tex]
