Find k^(')(x) if k(x)=e^(x)((4)/(5)x^(3)+(3)/(4)x^(-2)).

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Define Functions: We need to find the derivative of the function k(x) = e^(x)((4/5)x^(3) + (3/4)x^(-2)). To do this, we will use the product rule, 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's denote the first function as f(x) = e^(x) and the second function as g(x) = (4/5)x^(3) + (3/4)x^(-2).Find Derivative of f(x): First, we find the derivative of f(x) = e^(x). The derivative of e^(x) with respect to x is e^(x). So, f'(x) = e^(x).Find Derivative of g(x): Next, we find the derivative of g(x) = (4/5)x^(3) + (3/4)x^(-2). We will use the power rule, which states that the derivative of x^n is n*x^(n-1). For the first term, the derivative of (4/5)x^(3) is (4/5)*3*x^(3-1) = (12/5)x^(2). For the second term, the derivative of (3/4)x^(-2) is (3/4)*(-2)*x^(-2-1) = -(3/2)x^(-3).Apply Product Rule: Now we can apply the product rule. The derivative of k(x), denoted as k'(x), is f'(x)g(x) + f(x)g'(x). Substituting the derivatives we found, we get k'(x) = e^(x)((4/5)x^(3) + (3/4)x^(-2)) + e^(x)((12/5)x^(2) - (3/2)x^(-3)).Simplify Expression: We can simplify the expression by factoring out e^(x) from both terms. This gives us k'(x) = e^(x)[((4/5)x^(3) + (3/4)x^(-2)) + ((12/5)x^(2) - (3/2)x^(-3))].Combine Like Terms: Finally, we combine like terms within the brackets. However, there are no like terms to combine, so the expression is already simplified. The final answer is k'(x) = e^(x)[((4/5)x^(3) + (3/4)x^(-2)) + ((12/5)x^(2) - (3/2)x^(-3))].

Find $k'(x)$ if $k(x)=e^{x}\left(\frac{4}{5}x^{3}+\frac{3}{4}x^{-2}\right)$ .

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Find k′(x)k'(x)k′(x) if k(x)=ex(45x3+34x−2)k(x)=e^{x}\left(\frac{4}{5}x^{3}+\frac{3}{4}x^{-2}\right)k(x)=ex(54​x3+43​x−2).

Find $k'(x)$ if $k(x)=e^{x}\left(\frac{4}{5}x^{3}+\frac{3}{4}x^{-2}\right)$ .