Find the derivative of the following function. y=3^(-9x^(4)+3x^(3)) Answer: y^(')=

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Find the derivative of the following function.  y=3^(-9x^(4)+3x^(3)) Answer:  y^(')=

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Identify Function Components: Identify the function and its components. We have y = 3^(-9x^(4) + 3x^(3)). This is an exponential function with base 3 and exponent -9x^4 + 3x^3.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 a^u where a is a constant and u is a function of x, and the inner function is u(x) = -9x^4 + 3x^3.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 ln(a) is the natural logarithm of a. So, the derivative of 3^u with respect to u is 3^u * ln(3).Differentiate Inner Function: Differentiate the inner function with respect to x. The inner function is u(x) = -9x^4 + 3x^3. Using the power rule, the derivative of -9x^4 is -36x^3, and the derivative of 3x^3 is 9x^2. So, the derivative of u(x) with respect to x is u'(x) = -36x^3 + 9x^2.Apply Chain Rule Multiplication: Apply the chain rule by multiplying the derivatives from Step 3 and Step 4. Using the chain rule, the derivative of y with respect to x is y' = (3^u * ln(3)) * u'(x), where u = -9x^4 + 3x^3 and u'(x) = -36x^3 + 9x^2. Substituting u and u'(x) into the equation, we get y' = (3^(-9x^4 + 3x^3) * ln(3)) * (-36x^3 + 9x^2).Simplify Derivative Expression: Simplify the expression for the derivative. y' = 3^(-9x^4 + 3x^3) * ln(3) * (-36x^3 + 9x^2) can be simplified by factoring out common terms if possible. However, in this case, the expression is already in its simplest form.

Find the derivative of the following function. $\newline$ $y=3^{-9 x^{4}+3 x^{3}}$ $\newline$ Answer: $y^{\prime}=$

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Find the derivative of the following function.\newliney=3−9x4+3x3 y=3^{-9 x^{4}+3 x^{3}} y=3−9x4+3x3\newlineAnswer: y′= y^{\prime}= y′=

Find the derivative of the following function. $\newline$ $y=3^{-9 x^{4}+3 x^{3}}$ $\newline$ Answer: $y^{\prime}=$