Find k^(')(x) if k(x)=e^(x)(-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//2))". "

Question

Find  k^(')(x) if  k(x)=e^(x)(-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//2))". "

Bytelearn · Accepted Answer

Define Functions: We need to find the derivative of the function k(x) = e^(x)(-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//2)). To do this, we will use 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's denote the first function as f(x) = e^(x) and the second function as g(x) = (-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//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) = (d)/(dx)(e^(x)) = e^(x).Find Derivative of g(x): Next, we find the derivative of g(x) = (-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//2)). We will use the power rule for differentiation, which states that the derivative of x^n with respect to x is n*x^(n-1).Calculate g'(x): The derivative of the first term -(3)/(5)x^(-4//5) is: g1'(x) = (d)/(dx)(-(3)/(5)x^(-4//5)) = -(3)/(5)*(-4//5)*x^(-4//5-1) = (12)/(25)*x^(-9//5).Apply Product Rule: The derivative of the second term (1)/(5)x^(-3//2) is: g2'(x) = (d)/(dx)((1)/(5)x^(-3//2)) = (1)/(5)*(-3//2)*x^(-3//2-1) = -(3)/(10)*x^(-5//2).Simplify Expression: Now we add the derivatives of the two terms to get g'(x): g'(x) = g1'(x) + g2'(x) = (12)/(25)*x^(-9//5) - (3)/(10)*x^(-5//2).Combine Like Terms: 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 and functions we found earlier, we get: k'(x) = e^(x)(-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//2)) + e^(x)((12)/(25)*x^(-9//5) - (3)/(10)*x^(-5//2)).Final Derivative: We can simplify the expression by factoring out e^(x): k'(x) = e^(x)((-(3)/(5)x^(-4//5)+(1)/(5)x^(-3//2)) + (12)/(25)*x^(-9//5) - (3)/(10)*x^(-5//2)).Now we combine like terms: k'(x) = e^(x)((12)/(25)*x^(-9//5) - (3)/(5)*x^(-4//5) - (3)/(10)*x^(-5//2) + (1)/(5)*x^(-3//2)).This is the final simplified form of the derivative of k(x): k'(x) = e^(x)((12)/(25)*x^(-9//5) - (3)/(5)*x^(-4//5) - (3)/(10)*x^(-5//2) + (1)/(5)*x^(-3//2)).

Find $k'(x)$ if $\newline$ $k(x)=e^{x}\left(-\frac{3}{5}x^{-\frac{4}{5}}+\frac{1}{5}x^{-\frac{3}{2}}\right)$ .

Full solution

More problems from Find derivatives of using multiple formulae

Related topics

Find k′(x)k'(x)k′(x) if\newlinek(x)=ex(−35x−45+15x−32)k(x)=e^{x}\left(-\frac{3}{5}x^{-\frac{4}{5}}+\frac{1}{5}x^{-\frac{3}{2}}\right)k(x)=ex(−53​x−54​+51​x−23​).

Find $k'(x)$ if $\newline$ $k(x)=e^{x}\left(-\frac{3}{5}x^{-\frac{4}{5}}+\frac{1}{5}x^{-\frac{3}{2}}\right)$ .