find the derivative of the function. y = cos(a^8 + x^8) y(x) = need help? read it watch it submit answer

Explanation:

Step1: Apply chain - rule

Let $u = a^{8}+x^{8}$, then $y=\cos(u)$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

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

The derivative of $y = \cos(u)$ with respect to $u$ is $\frac{dy}{du}=-\sin(u)$.

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

Since $u = a^{8}+x^{8}$, and $a^{8}$ is a constant, the derivative of $u$ with respect to $x$ is $\frac{du}{dx}=8x^{7}$.

Step4: Calculate $\frac{dy}{dx}$

Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=-\sin(u)\cdot8x^{7}$. Replace $u$ with $a^{8}+x^{8}$, we get $y^{\prime}(x)=- 8x^{7}\sin(a^{8}+x^{8})$.

Answer:

$-8x^{7}\sin(a^{8}+x^{8})$