Find the derivative of the following function. y=5^(8x^(2)) Answer: y^(')=

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Find the derivative of the following function.  y=5^(8x^(2)) Answer:  y^(')=

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Identify Components: Identify the components of the function. The function y = 5^(8x^(2)) is an exponential function where the base is a constant (5) and the exponent is a function of x (8x^(2)).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 5^u and the inner function is u = 8x^(2).Find Derivative Outer: Find the derivative of the outer function with respect to u. The derivative of a^u with respect to u, where a is a constant, is a^u * ln(a). Therefore, the derivative of 5^u with respect to u is 5^u * ln(5).Find Derivative Inner: Find the derivative of the inner function with respect to x. The inner function is u = 8x^(2). The derivative of 8x^(2) with respect to x is 16x, since d/dx(x^(2)) = 2x and we have a constant multiple of 8.Apply Chain Rule: Apply the chain rule using the derivatives from steps 3 and 4. 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' = (5^(8x^(2)) * ln(5)) * (16x).Simplify Derivative: Simplify the expression for the derivative. y' = 16x * 5^(8x^(2)) * ln(5). This is the simplified form of the derivative of the given function.

Find the derivative of the following function. $\newline$ $y=5^{8 x^{2}}$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function. $\newline$ $y=5^{8 x^{2}}$ $\newline$ Answer: $y^{\prime}=$