Solve x^-1/4x^4

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

Evaluate

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

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

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

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

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

Use the Commutative Property of Multiplication.

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

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

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

Multiply 4 times -1.

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

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

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

Add the exponents -1 and -4.

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

Raise 4 to the power -1.

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

Differentiate w.r.t. x

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

Solution Steps

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

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

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{-5})$$

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}.

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

Do the arithmetic.

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