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

Question

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

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Product Rule Application: To find the derivative of the function k(x) = e^(x)(-x^(3/5) - x^(-5/4)), we will use the product 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.Identifying Functions: First, let's identify the two functions that we are dealing with. We have f(x) = e^(x) and g(x) = -x^(3/5) - x^(-5/4). We will need to find f'(x) and g'(x).Derivative of f(x): The derivative of f(x) = e^(x) with respect to x is f'(x) = e^(x), because the derivative of e^(x) is e^(x).Derivative of g(x): Now we need to find the derivative of g(x) = -x^(3/5) - x^(-5/4). We will use the power rule, which states that the derivative of x^n with respect to x is n*x^(n-1).Combining Derivatives: The derivative of -x^(3/5) is (-3/5)*x^(3/5 - 1) = (-3/5)*x^(-2/5).Using Product Rule: The derivative of -x^(-5/4) is (-5/4)*x^(-5/4 - 1) = (-5/4)*x^(-9/4).Simplifying Expression: Now we can combine the derivatives of the two terms to find g'(x): g'(x) = (-3/5)*x^(-2/5) - (5/4)*x^(-9/4).Final Derivative: Using the product rule, the derivative of k(x) = f(x)g(x) is k'(x) = f'(x)g(x) + f(x)g'(x). Substituting the derivatives we found, we get: k'(x) = e^(x)(-x^(3/5) - x^(-5/4)) + e^(x)((-3/5)*x^(-2/5) - (5/4)*x^(-9/4)).We can simplify this expression by factoring out e^(x): k'(x) = e^(x)(-x^(3/5) - x^(-5/4) - (3/5)*x^(-2/5) - (5/4)*x^(-9/4)).This is the final derivative of the function k(x). We have successfully applied the product rule and the power rule to find the derivative.

Find $k'(x)$ if $k(x)=e^{x}(-x^{\frac{3}{5}}-x^{-\frac{5}{4}})$

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Find $k'(x)$ if $k(x)=e^{x}(-x^{\frac{3}{5}}-x^{-\frac{5}{4}})$