calculate the derivative of x^5-x+2/x^3+7
I will assume you meant
y = (x^5 - x + 2)/(x^3 + 7)
use the quotient rule
dy/dx = [(x^3 + 7)(5x^4 - 1) - (x^5 - x + 2)(3x^2 ]/(x^3+7)^2
I will leave the simplification up to you
To calculate the derivative of a function, we can use the rules of differentiation. The general rules we will need are:
1. The derivative of a constant is zero.
2. The Power rule: if we have a term of the form x^n, then the derivative of this term is n times x^(n-1).
3. The derivative of a sum or difference of terms is the sum or difference of their derivatives.
Now, let's differentiate the given function step by step:
1. Begin by applying the Power rule to the numerator:
The derivative of x^5 is 5x^(5-1) = 5x^4.
The derivative of -x is -1.
Therefore, the numerator becomes: 5x^4 - 1.
2. Next, apply the Power rule to the denominator:
The derivative of x^3 is 3x^(3-1) = 3x^2.
The derivative of 7 is zero since it is a constant.
Therefore, the denominator becomes: 3x^2.
3. Applying the rules of differentiation for a quotient, we get:
(numerator derivative * denominator - numerator * denominator derivative) / (denominator)^2
Plugging in the values, we obtain:
[(5x^4 - 1) * (3x^2) - (x^5 - x + 2) * (2 * x^3)] / (3x^2)^2
4. Simplify the expression by expanding and collecting like terms:
(15x^6 - 3x^2 - 2x^8 + 2x^4 + 4x^3) / (9x^4)
5. Finally, we can simplify the expression further if desired, but this is the derivative of the given function.
Therefore, the derivative of x^5 - x + 2/x^3 + 7 is:
(15x^6 - 3x^2 - 2x^8 + 2x^4 + 4x^3) / (9x^4)