Question

let (f(x)=3sin(sin(x^{6}))). then (f(x)=) question help: video

Explanation:

Step1: Apply chain - rule

Let $u = \sin(x^{6})$, then $f(x)=3\sin(u)$. The derivative of $\sin(u)$ with respect to $u$ is $\cos(u)$, and by the chain - rule, $\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}$. So, $\frac{df}{du}=3\cos(u)=3\cos(\sin(x^{6}))$.

Step2: Differentiate $u = \sin(x^{6})$

Let $v = x^{6}$. Then $u=\sin(v)$. The derivative of $\sin(v)$ with respect to $v$ is $\cos(v)$, and $\frac{du}{dv}=\cos(v)=\cos(x^{6})$. Also, the derivative of $v = x^{6}$ with respect to $x$ is $\frac{dv}{dx}=6x^{5}$.

Step3: Apply chain - rule again for $\frac{du}{dx}$

By the chain - rule, $\frac{du}{dx}=\frac{du}{dv}\cdot\frac{dv}{dx}$. So, $\frac{du}{dx}=\cos(x^{6})\cdot6x^{5}$.

Step4: Calculate $f^{\prime}(x)$

Since $f^{\prime}(x)=\frac{df}{du}\cdot\frac{du}{dx}$, substituting the values we found: $f^{\prime}(x)=3\cos(\sin(x^{6}))\cdot\cos(x^{6})\cdot6x^{5}=18x^{5}\cos(x^{6})\cos(\sin(x^{6}))$.

Answer:

$18x^{5}\cos(x^{6})\cos(\sin(x^{6}))$