Solve 15d^2/3d^4

$$\frac { 15 d ^ { 2 } } { 3 d ^ { 4 } }$$

Evaluate

$$\frac{5}{d^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(15d^{2}\right)^{1}\times \frac{1}{3d^{4}}$$

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

$$15^{1}\left(d^{2}\right)^{1}\times \frac{1}{3}\times \frac{1}{d^{4}}$$

Use the Commutative Property of Multiplication.

$$15^{1}\times \frac{1}{3}\left(d^{2}\right)^{1}\times \frac{1}{d^{4}}$$

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

$$15^{1}\times \frac{1}{3}d^{2}d^{4\left(-1\right)}$$

Multiply 4 times -1.

$$15^{1}\times \frac{1}{3}d^{2}d^{-4}$$

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

$$15^{1}\times \frac{1}{3}d^{2-4}$$

Add the exponents 2 and -4.

$$15^{1}\times \frac{1}{3}d^{-2}$$

Raise 15 to the power 1.

$$15\times \frac{1}{3}d^{-2}$$

Multiply 15 times \frac{1}{3}.

$$5d^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{15^{1}d^{2}}{3^{1}d^{4}}$$

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

$$\frac{15^{1}d^{2-4}}{3^{1}}$$

Subtract 4 from 2.

$$\frac{15^{1}d^{-2}}{3^{1}}$$

Divide 15 by 3.

$$5d^{-2}$$

Differentiate w.r.t. d

$$-\frac{10}{d^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{15}{3}d^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}d}(5d^{-2})$$

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}.

$$-2\times 5d^{-2-1}$$

Do the arithmetic.

$$-10d^{-3}$$