Find the derivative of the following function. y=7^(3x^(5)-2x^(4)) Answer: y^(')=

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Find the derivative of the following function.  y=7^(3x^(5)-2x^(4)) Answer:  y^(')=

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Identify Function Components: Identify the function and its components. 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. 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. The derivative of a^u with respect to u is a^u * 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 * 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. 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)) * ln(7)) * (15x^(4) - 8x^(3)).Simplify Expression: Simplify the expression. We can leave the derivative in its factored form, as y' = (7^(3x^(5)-2x^(4)) * ln(7)) * (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.

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