Solve 49x^2/7x^7

$$\frac { 49 x ^ { 2 } } { 7 x ^ { 7 } }$$

Evaluate

$$\frac{7}{x^{5}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(49x^{2}\right)^{1}\times \frac{1}{7x^{7}}$$

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

$$49^{1}\left(x^{2}\right)^{1}\times \frac{1}{7}\times \frac{1}{x^{7}}$$

Use the Commutative Property of Multiplication.

$$49^{1}\times \frac{1}{7}\left(x^{2}\right)^{1}\times \frac{1}{x^{7}}$$

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

$$49^{1}\times \frac{1}{7}x^{2}x^{7\left(-1\right)}$$

Multiply 7 times -1.

$$49^{1}\times \frac{1}{7}x^{2}x^{-7}$$

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

$$49^{1}\times \frac{1}{7}x^{2-7}$$

Add the exponents 2 and -7.

$$49^{1}\times \frac{1}{7}x^{-5}$$

Raise 49 to the power 1.

$$49\times \frac{1}{7}x^{-5}$$

Multiply 49 times \frac{1}{7}.

$$7x^{-5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{49^{1}x^{2}}{7^{1}x^{7}}$$

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

$$\frac{49^{1}x^{2-7}}{7^{1}}$$

Subtract 7 from 2.

$$\frac{49^{1}x^{-5}}{7^{1}}$$

Divide 49 by 7.

$$7x^{-5}$$

Differentiate w.r.t. x

$$-\frac{35}{x^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{49}{7}x^{2-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(7x^{-5})$$

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}.

$$-5\times 7x^{-5-1}$$

Do the arithmetic.

$$-35x^{-6}$$