Solve 45/x/35x^6

$$\frac { \frac { 45 } { x } } { 35 x ^ { 6 } }$$

Evaluate

$$\frac{9}{7x^{7}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(45\times \frac{1}{x}\right)^{1}\times \frac{1}{35x^{6}}$$

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

$$45^{1}\times \left(\frac{1}{x}\right)^{1}\times \frac{1}{35}\times \frac{1}{x^{6}}$$

Use the Commutative Property of Multiplication.

$$45^{1}\times \frac{1}{35}\times \left(\frac{1}{x}\right)^{1}\times \frac{1}{x^{6}}$$

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

$$45^{1}\times \frac{1}{35}\times \frac{1}{x}x^{6\left(-1\right)}$$

Multiply 6 times -1.

$$45^{1}\times \frac{1}{35}\times \frac{1}{x}x^{-6}$$

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

$$45^{1}\times \frac{1}{35}x^{-1-6}$$

Add the exponents -1 and -6.

$$45^{1}\times \frac{1}{35}x^{-7}$$

Raise 45 to the power 1.

$$45\times \frac{1}{35}x^{-7}$$

Multiply 45 times \frac{1}{35}.

$$\frac{9}{7}x^{-7}$$

Differentiate w.r.t. x

$$-\frac{9}{x^{8}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{45}{35}x^{-1-6})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9}{7}x^{-7})$$

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}.

$$-7\times \frac{9}{7}x^{-7-1}$$

Do the arithmetic.

$$-9x^{-8}$$