Solve 6x^5/-4x^2

$$\frac{ 6 { x }^{ 5 } }{ -4 { x }^{ 2 } }$$

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(6x^{5}\right)^{1}\times \frac{1}{-4x^{2}}$$

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

$$6^{1}\left(x^{5}\right)^{1}\times \frac{1}{-4}\times \frac{1}{x^{2}}$$

Use the Commutative Property of Multiplication.

$$6^{1}\times \frac{1}{-4}\left(x^{5}\right)^{1}\times \frac{1}{x^{2}}$$

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

$$6^{1}\times \frac{1}{-4}x^{5}x^{2\left(-1\right)}$$

Multiply 2 times -1.

$$6^{1}\times \frac{1}{-4}x^{5}x^{-2}$$

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

$$6^{1}\times \frac{1}{-4}x^{5-2}$$

Add the exponents 5 and -2.

$$6^{1}\times \frac{1}{-4}x^{3}$$

Raise 6 to the power 1.

$$6\times \frac{1}{-4}x^{3}$$

Raise -4 to the power -1.

$$6\left(-\frac{1}{4}\right)x^{3}$$

Multiply 6 times -\frac{1}{4}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{6^{1}x^{5}}{\left(-4\right)^{1}x^{2}}$$

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

$$\frac{6^{1}x^{5-2}}{\left(-4\right)^{1}}$$

Subtract 2 from 5.

$$\frac{6^{1}x^{3}}{\left(-4\right)^{1}}$$

Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.

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

Differentiate w.r.t. x

$$-\frac{9x^{2}}{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6}{-4}x^{5-2})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-\frac{9}{2}x^{2}$$