find the derivative of $z = \\tan(e^{-7w})$\n$\\frac{dz}{dw}=$

Explanation:

Step1: Apply chain - rule

Let $u = e^{-7w}$. Then $z=\tan(u)$. The chain - rule states that $\frac{dz}{dw}=\frac{dz}{du}\cdot\frac{du}{dw}$. First, find $\frac{dz}{du}$. The derivative of $y = \tan(u)$ with respect to $u$ is $\frac{dz}{du}=\sec^{2}(u)$.

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

The derivative of $u = e^{-7w}$ with respect to $w$ is $\frac{du}{dw}=-7e^{-7w}$ using the rule that the derivative of $e^{ax}$ with respect to $x$ is $ae^{ax}$.

Step3: Calculate $\frac{dz}{dw}$

Substitute $\frac{dz}{du}$ and $\frac{du}{dw}$ into the chain - rule formula: $\frac{dz}{dw}=\sec^{2}(e^{-7w})\cdot(-7e^{-7w})=-7e^{-7w}\sec^{2}(e^{-7w})$.

Answer:

$-7e^{-7w}\sec^{2}(e^{-7w})$