Fractional powers are when the exponent is a fraction or a decimal, such as 2^(1/2). Negative powers are when the exponent is a negative number, such as 2^(-3).
To differentiate a fractional power, use the power rule: multiply the coefficient by the original base raised to the power minus one. For example, to differentiate 3x^(1/2), the result would be 3(1/2)x^(-1/2) or 3/(2x^(1/2)).
The rule for differentiating negative powers is similar to the power rule for fractional powers. Multiply the coefficient by the original base raised to the power minus one. For example, to differentiate 4x^(-3), the result would be -12x^(-4).
Yes, negative powers can be written as fractions or decimals. For example, 2^(-3) can also be written as 2^(-3/1) or 2^(-0.5).
To differentiate expressions with both fractional and negative powers, use the power rule for each term separately. For example, to differentiate 5x^(1/2) + 3x^(-2), the result would be (5/2)x^(-1/2) + (-6)x^(-3).