Solve (2a)^-1*3a^2

$$( 2 a ) ^ { - 1 } \times 3 a ^ { 2 }$$

Evaluate

$$\frac{3a}{2}$$

Solution Steps

Expand \left(2a\right)^{-1}.

$$2^{-1}a^{-1}\times 3a^{2}$$

Calculate 2 to the power of -1 and get \frac{1}{2}.

$$\frac{1}{2}a^{-1}\times 3a^{2}$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{3}{2}a^{-1}a^{2}$$

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

$$\frac{3}{2}a^{1}$$

Calculate a to the power of 1 and get a.

$$\frac{3}{2}a$$

Differentiate w.r.t. a

$$\frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of a Derivative

Expand \left(2a\right)^{-1}.

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

Calculate 2 to the power of -1 and get \frac{1}{2}.

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{2}a^{-1}a^{2})$$

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

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

Calculate a to the power of 1 and get a.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{3}{2}a)$$

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

$$\frac{3}{2}a^{1-1}$$

Subtract 1 from 1.

$$\frac{3}{2}a^{0}$$

For any term t except 0, t^{0}=1.

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

For any term t, t\times 1=t and 1t=t.

$$\frac{3}{2}$$