Find the derivative of the following function. y=ln(9x^(6)+4x^(5)) Answer: y^(')=

Question

Find the derivative of the following function.  y=ln(9x^(6)+4x^(5)) Answer:  y^(')=

Bytelearn · Accepted Answer

Apply Chain Rule: step_1: Apply the chain rule to differentiate the natural logarithm function. The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this case, f(u) = ln(u) and g(x) = 9x^(6) + 4x^(5). The derivative of ln(u) with respect to u is 1/u, and we will need to find the derivative of g(x) with respect to x.Differentiate Inner Function: step_2: Differentiate the inner function g(x) = 9x^(6) + 4x^(5). Using the power rule, the derivative of x^n with respect to x is n*x^(n-1). Therefore, g'(x) = d/dx(9x^(6)) + d/dx(4x^(5)) = 54x^(5) + 20x^(4).Combine Derivatives: step_3: Combine the derivatives using the chain rule. Now we apply the chain rule: y' = f'(g(x)) * g'(x) = (1/(9x^(6) + 4x^(5))) * (54x^(5) + 20x^(4)).Simplify Expression: step_4: Simplify the expression. We can factor out an x^4 from the terms in g'(x) to get y' = (1/(9x^(6) + 4x^(5))) * x^(4) * (54x + 20).Further Simplify: step_5: Further simplify the expression. Now we can cancel out an x^4 from the numerator and denominator to get y' = (54x + 20)/(9x^(2) + 4x).

Find the derivative of the following function. $\newline$ $y=\ln \left(9 x^{6}+4 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$

Full solution

More problems from Find derivatives of inverse trigonometric functions

Related topics

Find the derivative of the following function.\newliney=ln⁡(9x6+4x5) y=\ln \left(9 x^{6}+4 x^{5}\right) y=ln(9x6+4x5)\newlineAnswer: y′= y^{\prime}= y′=

Find the derivative of the following function. $\newline$ $y=\ln \left(9 x^{6}+4 x^{5}\right)$ $\newline$ Answer: $y^{\prime}=$