Given the function y=-(5)/(4root(5)(x)), find (dy)/(dx). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer: (dy)/(dx)=

Question

Given the function  y=-(5)/(4root(5)(x)), find  (dy)/(dx). Express your answer in radical form without using negative exponents, simplifying all fractions. Answer:  (dy)/(dx)=

Bytelearn · Accepted Answer

Write Function & Differentiate: Write down the function and differentiate it using the chain rule. The function is y = -(5)/(4√(5x)). To differentiate this function with respect to x, 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.Rewrite for Easier Differentiation: Rewrite the function in a form that is easier to differentiate. The function y = -(5)/(4√(5x)) can be rewritten as y = -(5)/(4*(5x)^(1/2)). This is because the square root of a number is the same as raising that number to the power of 1/2.Differentiate with Power Rule: Differentiate the function with respect to x. Using the power rule, which states that the derivative of x^n with respect to x is n*x^(n-1), we differentiate the function. The power rule is applied to the term (5x)^(1/2), and we also need to apply the constant multiple rule, which allows us to pull constants out of the derivative. (dy)/(dx) = -(5/4) * d/dx[(5x)^(1/2)] (dy)/(dx) = -(5/4) * (1/2) * (5x)^(-1/2) * d/dx[5x]Differentiate Inner Function: Differentiate the inner function 5x with respect to x. The derivative of 5x with respect to x is simply 5, because the derivative of x is 1 and the 5 is a constant coefficient. (dy)/(dx) = -(5/4) * (1/2) * (5x)^(-1/2) * 5Simplify Expression: Simplify the expression. Now we multiply the constants and simplify the expression. (dy)/(dx) = -(5/4) * (1/2) * 5 * (5x)^(-1/2) (dy)/(dx) = -(25/8) * (5x)^(-1/2)Express Derivative in Radical Form: Express the derivative in radical form without using negative exponents. To express (5x)^(-1/2) in radical form, we rewrite it as 1/√(5x). (dy)/(dx) = -(25/8) * (1/√(5x))Simplify Fraction: Simplify the fraction if possible. In this case, the fraction is already simplified, and we cannot simplify it further without changing the form requested in the problem. (dy)/(dx) = -(25/8) * (1/√(5x))

Given the function $y=-\frac{5}{4 \sqrt[5]{x}}$ , find $\frac{d y}{d x}$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $\frac{d y}{d x}=$

Full solution

More problems from Evaluate functions

Related topics

Given the function y=−54x5 y=-\frac{5}{4 \sqrt[5]{x}} y=−45x​5​, find dydx \frac{d y}{d x} dxdy​. Express your answer in radical form without using negative exponents, simplifying all fractions.\newlineAnswer: dydx= \frac{d y}{d x}= dxdy​=

Given the function $y=-\frac{5}{4 \sqrt[5]{x}}$ , find $\frac{d y}{d x}$ . Express your answer in radical form without using negative exponents, simplifying all fractions. $\newline$ Answer: $\frac{d y}{d x}=$