Solve 2/3x^2/frac{4{28}x^4}

$$\frac { \frac { 2 } { 3 } x ^ { 2 } } { \frac { 4 } { 28 } x ^ { 4 } }$$

Evaluate

$$\frac{14}{3x^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(\frac{2}{3}x^{2}\right)^{1}\times \frac{1}{\frac{1}{7}x^{4}}$$

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

$$\left(\frac{2}{3}\right)^{1}\left(x^{2}\right)^{1}\times \frac{1}{\frac{1}{7}}\times \frac{1}{x^{4}}$$

Use the Commutative Property of Multiplication.

$$\left(\frac{2}{3}\right)^{1}\times \frac{1}{\frac{1}{7}}\left(x^{2}\right)^{1}\times \frac{1}{x^{4}}$$

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

$$\left(\frac{2}{3}\right)^{1}\times \frac{1}{\frac{1}{7}}x^{2}x^{4\left(-1\right)}$$

Multiply 4 times -1.

$$\left(\frac{2}{3}\right)^{1}\times \frac{1}{\frac{1}{7}}x^{2}x^{-4}$$

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

$$\left(\frac{2}{3}\right)^{1}\times \frac{1}{\frac{1}{7}}x^{2-4}$$

Add the exponents 2 and -4.

$$\left(\frac{2}{3}\right)^{1}\times \frac{1}{\frac{1}{7}}x^{-2}$$

Raise \frac{2}{3} to the power 1.

$$\frac{2}{3}\times \frac{1}{\frac{1}{7}}x^{-2}$$

Raise \frac{1}{7} to the power -1.

$$\frac{2}{3}\times 7x^{-2}$$

Multiply \frac{2}{3} times 7.

$$\frac{14}{3}x^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{\left(\frac{2}{3}\right)^{1}x^{2}}{\left(\frac{1}{7}\right)^{1}x^{4}}$$

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

$$\frac{\left(\frac{2}{3}\right)^{1}x^{2-4}}{\left(\frac{1}{7}\right)^{1}}$$

Subtract 4 from 2.

$$\frac{\left(\frac{2}{3}\right)^{1}x^{-2}}{\left(\frac{1}{7}\right)^{1}}$$

Divide \frac{2}{3} by \frac{1}{7} by multiplying \frac{2}{3} by the reciprocal of \frac{1}{7}.

$$\frac{14}{3}x^{-2}$$

Differentiate w.r.t. x

$$-\frac{28}{3x^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{2}{3}}{\frac{1}{7}}x^{2-4})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{14}{3}x^{-2})$$

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}.

$$-2\times \frac{14}{3}x^{-2-1}$$

Do the arithmetic.

$$-\frac{28}{3}x^{-3}$$