Solve (a^1/-3a^3

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(a^{1}\right)^{1}\times \frac{1}{-3a^{3}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$1^{1}\left(a^{1}\right)^{1}\times \frac{1}{-3}\times \frac{1}{a^{3}}$$

Use the Commutative Property of Multiplication.

$$1^{1}\times \frac{1}{-3}\left(a^{1}\right)^{1}\times \frac{1}{a^{3}}$$

To raise a power to another power, multiply the exponents.

$$1^{1}\times \frac{1}{-3}a^{1}a^{3\left(-1\right)}$$

Multiply 3 times -1.

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

To multiply powers of the same base, add their exponents.

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

Add the exponents 1 and -3.

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

Raise -3 to the power -1.

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

Differentiate w.r.t. a

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

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Do the arithmetic.

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-2\left(-\frac{1}{3}\right)a^{-2-1}$$

Do the arithmetic.

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