Answer :

Sure! Let's find the derivative of the function [tex]\( y = e^{-7x^6 + 8x^5} \)[/tex].

### Step 1: Identify the Function
The function we have is an exponential function:
[tex]\[ y = e^{-7x^6 + 8x^5} \][/tex]

### Step 2: Apply the Chain Rule
To differentiate this, we need to use the chain rule. The chain rule states that if you have a composite function [tex]\( y = e^{u(x)} \)[/tex], then the derivative is:
[tex]\[ \frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx} \][/tex]

In our function, [tex]\( u(x) = -7x^6 + 8x^5 \)[/tex].

### Step 3: Differentiate [tex]\( u(x) \)[/tex]
Now we differentiate [tex]\( u(x) = -7x^6 + 8x^5 \)[/tex]:
- The derivative of [tex]\(-7x^6\)[/tex] is [tex]\(-42x^5\)[/tex].
- The derivative of [tex]\(8x^5\)[/tex] is [tex]\(40x^4\)[/tex].

So, the derivative of [tex]\( u(x) \)[/tex] is:
[tex]\[ \frac{du}{dx} = -42x^5 + 40x^4 \][/tex]

### Step 4: Put It All Together
Using the chain rule, we find the derivative of [tex]\( y \)[/tex]:
[tex]\[ \frac{dy}{dx} = e^{-7x^6 + 8x^5} \cdot (-42x^5 + 40x^4) \][/tex]

Which simplifies to:
[tex]\[ \frac{dy}{dx} = (-42x^5 + 40x^4) \cdot e^{-7x^6 + 8x^5} \][/tex]

And that's the derivative of the function!

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