Find (d)/(dx)(-sin(4x+1)) Answer:

Identify Functions: We are asked to find the derivative of the function -sin(4x+1) with respect to x. To do this, we will use the chain rule, which 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.Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is -sin(u), and the inner function is u = 4x+1. We will take the derivative of the outer function with respect to u, and then multiply it by the derivative of the inner function with respect to x.Derivative of Inner Function: The derivative of -sin(u) with respect to u is -cos(u), according to the derivative rule for sine, which states that (d/du)(sin(u)) = cos(u). Since we have a negative sign in front of sin(u), it becomes -cos(u).Apply Chain Rule: Now, we need to find the derivative of the inner function u = 4x+1 with respect to x. The derivative of 4x with respect to x is 4, and the derivative of a constant (1 in this case) is 0. So, the derivative of u with respect to x is 4.Simplify Final Answer: We can now apply the chain rule. The derivative of -sin(4x+1) with respect to x is the derivative of the outer function evaluated at the inner function (-cos(4x+1)) times the derivative of the inner function (4). So, we have: (d/dx)(-sin(4x+1)) = -cos(4x+1) * 4Simplify the expression to get the final answer. Multiplying -cos(4x+1) by 4 gives us: -4cos(4x+1) This is the derivative of -sin(4x+1) with respect to x.

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Q. Find

\frac{d}{d x}(-\sin (4 x+1))

\newline

Answer:

Identify Functions: We are asked to find the derivative of the function $-\sin(4x+1)$ with respect to $x$ . To do this, we will use the chain rule, which 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.
Derivative of Outer Function: First, let's identify the outer function and the inner function. The outer function is $-\sin(u)$ , and the inner function is $u = 4x+1$ . We will take the derivative of the outer function with respect to $u$ , and then multiply it by the derivative of the inner function with respect to $x$ .
Derivative of Inner Function: The derivative of $-\sin(u)$ with respect to $u$ is $-\cos(u)$ , according to the derivative rule for sine, which states that $\frac{d}{du}(\sin(u)) = \cos(u)$ . Since we have a negative sign in front of $\sin(u)$ , it becomes $-\cos(u)$ .
Apply Chain Rule: Now, we need to find the derivative of the inner function $u = 4x+1$ with respect to $x$ . The derivative of $4x$ with respect to $x$ is $4$ , and the derivative of a constant ( $1$ in this case) is $0$ . So, the derivative of $u$ with respect to $x$ is $4$ .
Simplify Final Answer: We can now apply the chain rule. The derivative of $-\sin(4x+1)$ with respect to $x$ is the derivative of the outer function evaluated at the inner function $-\cos(4x+1)$ times the derivative of the inner function $4$ . So, we have: $\newline$ $(d/dx)(-\sin(4x+1)) = -\cos(4x+1) \times 4$
Simplify Final Answer: We can now apply the chain rule. The derivative of $-\sin(4x+1)$ with respect to $x$ is the derivative of the outer function evaluated at the inner function $(-\cos(4x+1))$ times the derivative of the inner function $4$ . So, we have: $\newline$ $(\frac{d}{dx})(-\sin(4x+1)) = -\cos(4x+1) \times 4$ Simplify the expression to get the final answer. Multiplying $-\cos(4x+1)$ by $4$ gives us: $\newline$ $-4\cos(4x+1)$ $\newline$ This is the derivative of $-\sin(4x+1)$ with respect to $x$ .