Solve 4x^4/16x^7

$$\frac { 4 x ^ { 4 } } { 16 x ^ { 7 } }$$

Evaluate

$$\frac{1}{4x^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(4x^{4}\right)^{1}\times \frac{1}{16x^{7}}$$

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

$$4^{1}\left(x^{4}\right)^{1}\times \frac{1}{16}\times \frac{1}{x^{7}}$$

Use the Commutative Property of Multiplication.

$$4^{1}\times \frac{1}{16}\left(x^{4}\right)^{1}\times \frac{1}{x^{7}}$$

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

$$4^{1}\times \frac{1}{16}x^{4}x^{7\left(-1\right)}$$

Multiply 7 times -1.

$$4^{1}\times \frac{1}{16}x^{4}x^{-7}$$

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

$$4^{1}\times \frac{1}{16}x^{4-7}$$

Add the exponents 4 and -7.

$$4^{1}\times \frac{1}{16}x^{-3}$$

Raise 4 to the power 1.

$$4\times \frac{1}{16}x^{-3}$$

Multiply 4 times \frac{1}{16}.

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

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{4}}{16^{1}x^{7}}$$

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

$$\frac{4^{1}x^{4-7}}{16^{1}}$$

Subtract 7 from 4.

$$\frac{4^{1}x^{-3}}{16^{1}}$$

Reduce the fraction \frac{4}{16} to lowest terms by extracting and canceling out 4.

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

Differentiate w.r.t. x

$$-\frac{3}{4x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{16}x^{4-7})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$-\frac{3}{4}x^{-4}$$