Find k^(')(x) if k(x)=-3x*e^(-4x^(3)+3x^(2)).

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Identify Components: Identify the components of the function that will require the use of the product rule and the chain rule for differentiation. The function k(x) = -3x * e^(-4x^3 + 3x^2) is a product of two functions, -3x and e^(-4x^3 + 3x^2). The exponential function also contains a composite function, which will require the use of the chain rule.Apply Product Rule: Apply the product rule for differentiation, which 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. Let u = -3x and v = e^(-4x^3 + 3x^2). Then, k'(x) = u'v + uv'.Differentiate u: Differentiate u = -3x with respect to x to get u' = -3.Differentiate v: Differentiate v = e^(-4x^3 + 3x^2) with respect to x using 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. Let g(x) = -4x^3 + 3x^2 be the inner function. Then, v' = e^g(x) * g'(x).Differentiate g: Differentiate g(x) = -4x^3 + 3x^2 with respect to x to get g'(x) = -12x^2 + 6x.Substitute g' into v': Substitute g'(x) into the expression for v' to get v' = e^(-4x^3 + 3x^2) * (-12x^2 + 6x).Substitute into Product Rule: Substitute u', v, and v' into the product rule expression k'(x) = u'v + uv' to get k'(x) = -3 * e^(-4x^3 + 3x^2) + (-3x) * (e^(-4x^3 + 3x^2) * (-12x^2 + 6x)).Simplify Expression: Simplify the expression for k'(x) by distributing and combining like terms. k'(x) = -3e^(-4x^3 + 3x^2) - 3x * e^(-4x^3 + 3x^2) * (-12x^2 + 6x).Factor out Common Factor: Factor out the common factor e^(-4x^3 + 3x^2) to get k'(x) = e^(-4x^3 + 3x^2) * (-3 - 3x * (-12x^2 + 6x)).Distribute -3x: Continue simplifying the expression by distributing -3x inside the parentheses. k'(x) = e^(-4x^3 + 3x^2) * (-3 + 36x^3 - 18x^2).Combine Like Terms: Combine like terms inside the parentheses to get the final expression for k'(x). k'(x) = e^(-4x^3 + 3x^2) * (36x^3 - 18x^2 - 3).

Find $k^{\prime}(x)$ if $k(x)=-3 x \cdot e^{-4 x^{3}+3 x^{2}}$ .

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Find k′(x) k^{\prime}(x) k′(x) if k(x)=−3x⋅e−4x3+3x2 k(x)=-3 x \cdot e^{-4 x^{3}+3 x^{2}} k(x)=−3x⋅e−4x3+3x2.

Find $k^{\prime}(x)$ if $k(x)=-3 x \cdot e^{-4 x^{3}+3 x^{2}}$ .