find the derivative of the function. y = 7^5 - x^2 y = i need help? read it submit answer

Explanation:

Step1: Recall chain - rule

Let $u = 5 - x^{2}$, then $y = 7^{u}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.

Step2: Differentiate $y$ with respect to $u$

The derivative of $y = a^{u}$ (where $a = 7$) with respect to $u$ is $\frac{dy}{du}=7^{u}\ln(7)$ (since the derivative of $a^{u}$ with respect to $u$ is $a^{u}\ln(a)$).

Step3: Differentiate $u$ with respect to $x$

Given $u = 5 - x^{2}$, then $\frac{du}{dx}=-2x$.

Step4: Apply the chain - rule

$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=7^{5 - x^{2}}\ln(7)\cdot(-2x)=- 2x\ln(7)\cdot7^{5 - x^{2}}$.

Answer:

$-2x\ln(7)\cdot7^{5 - x^{2}}$