Find the derivative of the following function. y=ln(5x^(2)+x) Answer: y^(')=

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Find the derivative of the following function.  y=ln(5x^(2)+x) Answer:  y^(')=

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Identify function: Identify the function to differentiate. The function given is y = ln(5x^2 + x). We need to find the derivative of this function with respect to x.Apply chain rule: Apply the chain rule for differentiation. 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 ln(u) and the inner function is u = 5x^2 + x.Differentiate outer function: Differentiate the outer function with respect to the inner function. The derivative of ln(u) with respect to u is 1/u. So, if y = ln(u), then dy/du = 1/u.Differentiate inner function: Differentiate the inner function with respect to x. The inner function is u = 5x^2 + x. The derivative of 5x^2 with respect to x is 10x, and the derivative of x with respect to x is 1. Therefore, du/dx = 10x + 1.Apply chain rule: Apply the chain rule to find the derivative of y with respect to x. Using the chain rule, dy/dx = (dy/du) * (du/dx). We have dy/du = 1/u and du/dx = 10x + 1. Substituting u = 5x^2 + x into dy/du, we get dy/dx = (1/(5x^2 + x)) * (10x + 1).Simplify derivative: Simplify the expression for the derivative. The derivative of y with respect to x is dy/dx = (10x + 1) / (5x^2 + x). This is the final simplified form of the derivative.

Find the derivative of the following function. $\newline$ $y=\ln \left(5 x^{2}+x\right)$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function.\newliney=ln⁡(5x2+x) y=\ln \left(5 x^{2}+x\right) y=ln(5x2+x)\newlineAnswer: y′= y^{\prime}= y′=

Find the derivative of the following function. $\newline$ $y=\ln \left(5 x^{2}+x\right)$ $\newline$ Answer: $y^{\prime}=$