Question

attempt 1: 10 attempts remaining. find the derivative of the function ( y = e^{(x^2 - 8)^6} ) using the chain rule for exponential functions. ( \frac{dy}{dx} = ) input box submit answer next item

Explanation:

Step1: Identify outer and inner functions

The function is \( y = e^{(x^2 - 8)^6} \). Let the outer function be \( f(u)=e^u \) and the inner function be \( u = (x^2 - 8)^6 \).

Step2: Differentiate outer function

The derivative of \( f(u)=e^u \) with respect to \( u \) is \( f'(u)=e^u \).

Step3: Differentiate inner function

Now, differentiate \( u = (x^2 - 8)^6 \). Let \( v=x^2 - 8 \), so \( u = v^6 \). First, differentiate \( u \) with respect to \( v \): \( \frac{du}{dv}=6v^5 \). Then differentiate \( v \) with respect to \( x \): \( \frac{dv}{dx}=2x \). By the chain rule, \( \frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}=6(x^2 - 8)^5\cdot2x = 12x(x^2 - 8)^5 \).

Step4: Apply chain rule to original function

By the chain rule, \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). We know \( \frac{dy}{du}=e^u = e^{(x^2 - 8)^6} \) and \( \frac{du}{dx}=12x(x^2 - 8)^5 \). So, \( \frac{dy}{dx}=e^{(x^2 - 8)^6}\cdot12x(x^2 - 8)^5 \).

Answer:

\( 12x(x^2 - 8)^5 e^{(x^2 - 8)^6} \)