Question

attempt 1: 10 attempts remaining. find the derivative of the function ( y = e^{(x^2 - 7)^3} ) using the chain rule for exponential functions. ( \frac{dy}{dx} = ) submit answer next item

Explanation:

Step1: Identify outer and inner functions

Let \( u=(x^{2}-7)^{3} \), so \( y = e^{u} \).

Step2: Differentiate outer function

The derivative of \( y = e^{u} \) with respect to \( u \) is \( \frac{dy}{du}=e^{u} \).

Step3: Differentiate inner function

Now, differentiate \( u=(x^{2}-7)^{3} \) with respect to \( x \). Let \( v = x^{2}-7 \), so \( u = v^{3} \). First, derivative of \( u \) with respect to \( v \) is \( \frac{du}{dv}=3v^{2} \), and derivative of \( v \) with respect to \( x \) is \( \frac{dv}{dx}=2x \). By chain rule, \( \frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}=3(x^{2}-7)^{2}\cdot2x = 6x(x^{2}-7)^{2} \).

Step4: Apply chain rule for \( y \) and \( x \)

By chain rule, \( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} \). Substitute \( \frac{dy}{du}=e^{u} \) (with \( u=(x^{2}-7)^{3} \)) and \( \frac{du}{dx}=6x(x^{2}-7)^{2} \), we get \( \frac{dy}{dx}=e^{(x^{2}-7)^{3}}\cdot6x(x^{2}-7)^{2} \).

Answer:

\( 6x(x^{2}-7)^{2}e^{(x^{2}-7)^{3}} \)