Solve 3n^-frac{1{2}}*3n

$$3 n ^ { - \frac { 1 } { 2 } } \cdot 3 n$$

Evaluate

$$9\sqrt{n}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -\frac{1}{2} and 1 to get \frac{1}{2}.

$$3n^{\frac{1}{2}}\times 3$$

Multiply 3 and 3 to get 9.

$$9n^{\frac{1}{2}}$$

Differentiate w.r.t. n

$$\frac{9}{2\sqrt{n}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add -\frac{1}{2} and 1 to get \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(3n^{\frac{1}{2}}\times 3)$$

Multiply 3 and 3 to get 9.

$$\frac{\mathrm{d}}{\mathrm{d}n}(9n^{\frac{1}{2}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{2}\times 9n^{\frac{1}{2}-1}$$

Multiply \frac{1}{2} times 9.

$$\frac{9}{2}n^{\frac{1}{2}-1}$$

Subtract 1 from \frac{1}{2}.

$$\frac{9}{2}n^{-\frac{1}{2}}$$