calculate the derivative of the following function.\ny = (x^2 + 8x - 9)^4\n\frac{dy}{dx}=square

Explanation:

Step1: Let $u = x^{2}+8x - 9$

$y = u^{4}$

Step2: Find $\frac{du}{dx}$

$\frac{du}{dx}=\frac{d}{dx}(x^{2}+8x - 9)=2x + 8$

Step3: Find $\frac{dy}{du}$

$\frac{dy}{du}=\frac{d}{du}(u^{4})=4u^{3}$

Step4: Use chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$

Substitute $u = x^{2}+8x - 9$ back in: $\frac{dy}{dx}=4(x^{2}+8x - 9)^{3}\cdot(2x + 8)$ Simplify: $\frac{dy}{dx}=4(2x + 8)(x^{2}+8x - 9)^{3}=8(x + 4)(x^{2}+8x - 9)^{3}$

Answer:

$8(x + 4)(x^{2}+8x - 9)^{3}$