Find the derivative. Express answer in exponential notation without any negative exponents.
y= 1/(sqroot5x+2)^3
This reads: 1 divided by the square root of (5x+2) to the third
To find the derivative of the function y = 1 / (√(5x + 2))^3, you can follow these steps:
Step 1: Rewrite the function using exponentiation notation.
y = (5x + 2)^(-3/2)
Step 2: Apply the power rule of differentiation, which states that when you have a function raised to a power, you can bring down the power as a coefficient and then reduce the power by 1.
For instance, if you have a function u(x) raised to the power of n, where n is a constant, the derivative is given by:
d/dx [u(x)^n] = n * u(x)^(n-1) * du(x)/dx
In our case, let u = 5x + 2, and n = -3/2. Therefore,
dy/dx = (-3/2) * (5x + 2)^(-3/2 - 1) * d/dx[5x + 2]
Step 3: Differentiate 5x + 2 with respect to x. The derivative of 5x is simply 5, and the derivative of 2 is 0 since it is a constant. Thus, d/dx[5x + 2] = 5.
Substituting this in, we have:
dy/dx = (-3/2) * (5x + 2)^(-3/2 - 1) * 5
Step 4: Simplify the expression by combining the terms and expressing the answer in exponential notation without negative exponents.
(-3/2) * (5x + 2)^(-3/2 - 1) * 5
= -15/2 * (5x + 2)^(-5/2) * 5
= -75/2 * (5x + 2)^(-5/2)
Thus, the derivative of y = 1 / (√(5x + 2))^3 is dy/dx = -75/2 * (5x + 2)^(-5/2).