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Given that
x=w^(5)+5, find
(d)/(dw)(3x^(2)+3cos w) in terms of only
w.
Answer:

Given that $x=w^{5}+5$ , find $\frac{d}{d w}\left(3 x^{2}+3 \cos w\right)$ in terms of only $w$ . $\newline$ Answer:

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Write a formula for an arithmetic sequence

Full solution

Q. Given that

x=w^{5}+5

, find

\frac{d}{d w}\left(3 x^{2}+3 \cos w\right)

in terms of only

w

.

\newline

Answer:

Express $x$ in terms of $w$ : First, we need to express $x$ in terms of $w$ to be able to differentiate the function with respect to $w$ . We are given that $x = w^5 + 5$ . We will use this to differentiate the function $3x^2 + 3\cos(w)$ with respect to $w$ .
Apply chain rule: Next, we need to apply the chain rule to differentiate $3x^2$ with respect to $w$ . The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $3u^2$ (where $u = x$ ) and the inner function is $x = w^5 + 5$ .
Derivative of $x$ with respect to $w$ : The derivative of $3u^2$ with respect to $u$ is $6u$ . So, the derivative of $3x^2$ with respect to $x$ is $6x$ . Now we need to multiply this by the derivative of $x$ with respect to $w$ , which is the derivative of $w$ $0$ with respect to $w$ .
Multiply derivatives: The derivative of $w^5$ with respect to $w$ is $5w^4$ , and the derivative of a constant $(5)$ is $0$ . Therefore, the derivative of $x$ with respect to $w$ is $5w^4$ .
Substitute $x$ into expression: Now we can multiply the derivative of $3x^2$ with respect to $x$ by the derivative of $x$ with respect to $w$ to get the derivative of $3x^2$ with respect to $w$ . This gives us $6x \times 5w^4$ .
Differentiate $3\cos(w)$ : Substituting $x = w^5 + 5$ into $6x \cdot 5w^4$ , we get $6(w^5 + 5) \cdot 5w^4$ . This simplifies to $30w^4(w^5 + 5)$ .
Combine derivatives: Now we need to differentiate $3\cos(w)$ with respect to $w$ . The derivative of $\cos(w)$ with respect to $w$ is $-\sin(w)$ . Therefore, the derivative of $3\cos(w)$ with respect to $w$ is $-3\sin(w)$ .
Combine derivatives: Now we need to differentiate $3\cos(w)$ with respect to $w$ . The derivative of $\cos(w)$ with respect to $w$ is $-\sin(w)$ . Therefore, the derivative of $3\cos(w)$ with respect to $w$ is $-3\sin(w)$ .Finally, we add the derivatives of $3x^2$ and $3\cos(w)$ with respect to $w$ to get the derivative of the entire function with respect to $w$ . This gives us $w$ $2$ .

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