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Find the derivative of the following function.

y=7^(3x^(5)-2x^(4))
Answer:
y^(')=

Find the derivative of the following function. $\newline$ $y=7^{3 x^{5}-2 x^{4}}$ $\newline$ Answer: $y^{\prime}=$

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Find derivatives of radical functions

Full solution

Q. Find the derivative of the following function.

\newline

y=7^{3 x^{5}-2 x^{4}}

\newline

Answer:

y^{\prime}=

Identify Function Components: Identify the function and its components. $\newline$ We have $y = 7^{3x^{5}-2x^{4}}$ . This is an exponential function with base $7$ and an exponent that is a polynomial in $x$ .
Apply Chain Rule: Apply the chain rule for derivatives. $\newline$ 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 $7^u$ and the inner function is $u(x) = 3x^{5}-2x^{4}$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $a^u$ with respect to $u$ is $a^u \cdot \ln(a)$ , where $a$ is a constant and $u$ is a function of $x$ . Therefore, the derivative of $7^u$ with respect to $u$ is $7^u \cdot \ln(7)$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The inner function $u(x) = 3x^{5}-2x^{4}$ is a polynomial, and we can differentiate it term by term. The derivative of $3x^{5}$ with respect to $x$ is $15x^{4}$ , and the derivative of $-2x^{4}$ with respect to $x$ is $-8x^{3}$ . Therefore, the derivative of $u(x)$ is $15x^{4} - 8x^{3}$ .
Combine Using Chain Rule: Combine the results using the chain rule. $\newline$ The derivative of $y$ with respect to $x$ is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This gives us $y' = (7^{3x^{5}-2x^{4}} \cdot \ln(7)) \cdot (15x^{4} - 8x^{3})$ .
Simplify Expression: Simplify the expression. $\newline$ We can leave the derivative in its factored form, as $y' = (7^{3x^{5}-2x^{4}} \cdot \ln(7)) \cdot (15x^{4} - 8x^{3})$ , or we can distribute the $\ln(7)$ to both terms in the parentheses, but it is not necessary for the final answer.

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