Solve 4x^2/4x^7

$$\frac { 4 x ^ { 2 } } { 4 x ^ { 7 } }$$

Evaluate

$$\frac{1}{x^{5}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 7 times -1.

$$4^{1}\times \frac{1}{4}x^{2}x^{-7}$$

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

$$4^{1}\times \frac{1}{4}x^{2-7}$$

Add the exponents 2 and -7.

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

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

$$4^{1-1}x^{-5}$$

Add the exponents 1 and -1.

$$4^{0}x^{-5}$$

For any term t except 0, t^{0}=1.

$$1x^{-5}$$

For any term t, t\times 1=t and 1t=t.

$$x^{-5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{2}}{4^{1}x^{7}}$$

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

$$4^{1-1}x^{2-7}$$

Subtract 1 from 1.

$$4^{0}x^{2-7}$$

For any number a except 0, a^{0}=1.

$$x^{2-7}$$

Subtract 7 from 2.

$$x^{-5}$$

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

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

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}.

$$-5x^{-5-1}$$

Do the arithmetic.

$$-5x^{-6}$$