Find the derivative of the following function. y=e^(8x^(6)+5x^(5)) Answer: y^(')=

Identify u as exponent: To find the derivative of the function y=e^(8x^(6)+5x^(5)), we will use the chain rule. The chain rule states that the derivative of e^u, where u is a function of x, is e^u times the derivative of u with respect to x.Find derivative of u: First, let's identify u as the exponent of e. In this case, u = 8x^(6) + 5x^(5).Apply power rule: Now we need to find the derivative of u with respect to x, which is u'. To do this, we apply the power rule to each term separately. The power rule states that the derivative of x^n is n*x^(n-1).Apply chain rule: The derivative of the first term, 8x^(6), is 48x^(5) because we multiply the exponent 6 by the coefficient 8 and then decrease the exponent by 1. The derivative of the second term, 5x^(5), is 25x^(4) because we multiply the exponent 5 by the coefficient 5 and then decrease the exponent by 1. So, u' = 48x^(5) + 25x^(4).Final derivative: Now we can apply the chain rule. The derivative of y with respect to x, y', is e^(u) times u'. So, y' = e^(8x^(6)+5x^(5)) * (48x^(5) + 25x^(4)).We have found the derivative of the function without any mathematical errors. The final answer is y' = e^(8x^(6)+5x^(5)) * (48x^(5) + 25x^(4)).

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Find the derivative of the following function.

y=e^(8x^(6)+5x^(5))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=e^{8 x^{6}+5 x^{5}}$ $\newline$ Answer: $y^{\prime}=$

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Calculus

Find derivatives using logarithmic differentiation

Full solution

Q. Find the derivative of the following function.

\newline

y=e^{8 x^{6}+5 x^{5}}

\newline

Answer:

y^{\prime}=

Identify $u$ as exponent: To find the derivative of the function $y=e^{8x^{6}+5x^{5}}$ , we will use the chain rule. The chain rule states that the derivative of $e^{u}$ , where $u$ is a function of $x$ , is $e^{u}$ times the derivative of $u$ with respect to $x$ .
Find derivative of $u$ : First, let's identify $u$ as the exponent of $e$ . In this case, $u = 8x^{6} + 5x^{5}$ .
Apply power rule: Now we need to find the derivative of $u$ with respect to $x$ , which is $u'$ . To do this, we apply the power rule to each term separately. The power rule states that the derivative of $x^n$ is $n*x^{(n-1)}$ .
Apply chain rule: The derivative of the first term, $8x^{6}$ , is $48x^{5}$ because we multiply the exponent $6$ by the coefficient $8$ and then decrease the exponent by $1$ . The derivative of the second term, $5x^{5}$ , is $25x^{4}$ because we multiply the exponent $5$ by the coefficient $5$ and then decrease the exponent by $1$ . So, $48x^{5}$ $0$ .
Final derivative: Now we can apply the chain rule. The derivative of $y$ with respect to $x$ , $y'$ , is $e^{u}$ times $u'$ . So, $y' = e^{8x^{6}+5x^{5}} \times (48x^{5} + 25x^{4})$ .
Final derivative: Now we can apply the chain rule. The derivative of $y$ with respect to $x$ , $y'$ , is $e^{u}$ times $u'$ . So, $y' = e^{8x^{6}+5x^{5}} \times (48x^{5} + 25x^{4})$ .We have found the derivative of the function without any mathematical errors. The final answer is $y' = e^{8x^{6}+5x^{5}} \times (48x^{5} + 25x^{4})$ .