differentiation of 1/x^2 + under root x^3 whole square
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We need to find the derivative of the function f of x equals one over x squared plus square root of x cubed, all squared. This is a composite function that requires careful application of differentiation rules. Let's visualize this function first to understand its behavior.
The first step is to rewrite the expression using exponential notation. We convert one over x squared to x to the negative two power, and square root of x cubed becomes x to the three halves power. This gives us the function f of x equals x to the negative two plus x to the three halves, all squared.
Now we expand the square using the binomial formula. We get x to the negative four plus two times x to the negative one half plus x cubed. Each term follows the exponent rules: when we square x to the negative two, we get x to the negative four. When we multiply x to the negative two by x to the three halves, we get x to the negative one half. And when we square x to the three halves, we get x cubed.
Now we apply the power rule to differentiate each term. For x to the negative four, we get negative four x to the negative five. For two x to the negative one half, we get negative x to the negative three halves. For x cubed, we get three x squared. Combining all terms, the final derivative is negative four x to the negative five minus x to the negative three halves plus three x squared.
We have successfully found the derivative of the given function. The final answer is three x squared minus x to the negative three halves minus four x to the negative five. This can also be written using fractional notation as three x squared minus one over x to the three halves minus four over x to the fifth power. We solved this by rewriting the expression with exponents, expanding the square, and applying the power rule to each term.