Find the derivative of f(x)=-4x(3^(x^(4))).

Question

Find the derivative of  f(x)=-4x(3^(x^(4))).

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Product Rule Explanation: To find the derivative of f(x) = -4x(3^(x^(4))), we will use the product rule and the chain rule. The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.Derivative of u(x): Let's denote the two functions as u(x) = -4x and v(x) = 3^(x^(4)). We will find the derivatives of u(x) and v(x) separately.Derivative of v(x): The derivative of u(x) = -4x with respect to x is u'(x) = -4, since the derivative of x with respect to x is 1.Chain Rule Application: To find the derivative of v(x) = 3^(x^(4)), we will use the chain rule. 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.Derivative of v(x): The outer function is g(y) = 3^y, and the inner function is h(x) = x^4. The derivative of the outer function with respect to y is g'(y) = 3^y * ln(3), and the derivative of the inner function with respect to x is h'(x) = 4x^3.Product Rule Application: Applying the chain rule, the derivative of v(x) is v'(x) = g'(h(x)) * h'(x) = 3^(x^(4)) * ln(3) * 4x^3.Simplify Expression: Now we can apply the product rule to find the derivative of f(x) = u(x)v(x). Using the product rule, f'(x) = u'(x)v(x) + u(x)v'(x).Factor Out Common Term: Substituting the derivatives we found, f'(x) = (-4)(3^(x^(4))) + (-4x)(3^(x^(4)) * ln(3) * 4x^3).Simplify the expression to combine like terms. f'(x) = -4 * 3^(x^(4)) - 16x^4 * 3^(x^(4)) * ln(3).Factor out the common term 3^(x^(4)) to get the final derivative. f'(x) = 3^(x^(4)) * (-4 - 16x^4 * ln(3)).

Find the derivative of $\newline$ $f(x)=-4x(3^{x^{4}}).$

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Find the derivative of \newlinef(x)=−4x(3x4).f(x)=-4x(3^{x^{4}}).f(x)=−4x(3x4).

Find the derivative of $\newline$ $f(x)=-4x(3^{x^{4}}).$