Find dy/dx: y=csc(2x^(4)+6)

Apply Chain Rule: To find the derivative of y with respect to x, we need to apply the chain rule. 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 csc(u) and the inner function is u = 2x^4 + 6.Derivative of Outer Function: First, we find the derivative of the outer function csc(u) with respect to u. The derivative of csc(u) is -csc(u)cot(u). We will substitute u back in later.Derivative of Inner Function: Next, we find the derivative of the inner function u = 2x^4 + 6 with respect to x. The derivative of 2x^4 with respect to x is 8x^3, and the derivative of 6 with respect to x is 0. So, the derivative of u with respect to x is 8x^3.Apply Chain Rule Again: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: dy/dx = -csc(u)cot(u) * du/dx = -csc(2x^4 + 6)cot(2x^4 + 6) * 8x^3Final Derivative: We have found the derivative of y with respect to x in terms of x. The derivative is: dy/dx = -8x^3 * csc(2x^4 + 6) * cot(2x^4 + 6)

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Algebra 2

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Q. Find

\frac{dy}{dx}: y=\csc(2x^{4}+6)

Apply Chain Rule: To find the derivative of $y$ with respect to $x$ , we need to apply the chain rule. 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 $\csc(u)$ and the inner function is $u = 2x^4 + 6$ .
Derivative of Outer Function: First, we find the derivative of the outer function $\csc(u)$ with respect to $u$ . The derivative of $\csc(u)$ is $-\csc(u)\cot(u)$ . We will substitute $u$ back in later.
Derivative of Inner Function: Next, we find the derivative of the inner function $u = 2x^4 + 6$ with respect to $x$ . The derivative of $2x^4$ with respect to $x$ is $8x^3$ , and the derivative of $6$ with respect to $x$ is $0$ . So, the derivative of $u$ with respect to $x$ is $8x^3$ .
Apply Chain Rule Again: Now we apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function. This gives us: $\newline$ $\frac{dy}{dx} = -\csc(u)\cot(u) \times \frac{du}{dx}$ $\newline$ $= -\csc(2x^4 + 6)\cot(2x^4 + 6) \times 8x^3$
Final Derivative: We have found the derivative of $y$ with respect to $x$ in terms of $x$ . The derivative is: $\newline$ $\frac{dy}{dx} = -8x^3 \cdot \csc(2x^4 + 6) \cdot \cot(2x^4 + 6)$