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Find the derivative of the following function.

y=ln(9x^(6)+4x^(5))
Answer:
y^(')=

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

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Find derivatives of inverse trigonometric functions

Full solution

Q. Find the derivative of the following function.

\newline

y=\ln \left(9 x^{6}+4 x^{5}\right)

\newline

Answer:

y^{\prime}=

Apply Chain Rule: step_1: Apply the chain rule to differentiate the natural logarithm function. $\newline$ The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) \cdot g'(x)$ . 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 $n\cdot x^{(n-1)}$ . Therefore, $g'(x) = \frac{d}{dx}(9x^{6}) + \frac{d}{dx}(4x^{5}) = 54x^{5} + 20x^{4}$ .
Combine Derivatives: step_3: Combine the derivatives using the chain rule. $\newline$ Now we apply the chain rule: $y' = f'(g(x)) \cdot g'(x) = \frac{1}{9x^{6} + 4x^{5}} \cdot (54x^{5} + 20x^{4})$ .
Simplify Expression: step_4: Simplify the expression. $\newline$ We can factor out an $x^4$ from the terms in $g'(x)$ to get $y' = \frac{1}{9x^{6} + 4x^{5}} \cdot x^{4} \cdot (54x + 20)$ .
Further Simplify: step_5: Further simplify the expression. $\newline$ Now we can cancel out an $x^4$ from the numerator and denominator to get $y' = \frac{54x + 20}{9x^{2} + 4x}$ .

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