Solve (22+x/(4x)^2)^prime

$$( \frac { 22 + x } { ( 4 x ) ^ { 2 } } ) ^ { \prime }$$

Evaluate

$$-\frac{x+44}{16x^{3}}$$

Steps Using Derivative Rule for Quotient

Expand \left(4x\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{22+x}{4^{2}x^{2}})$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{22+x}{16x^{2}})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{16x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}+22)-\left(x^{1}+22\right)\frac{\mathrm{d}}{\mathrm{d}x}(16x^{2})}{\left(16x^{2}\right)^{2}}$$

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}.

$$\frac{16x^{2}x^{1-1}-\left(x^{1}+22\right)\times 2\times 16x^{2-1}}{\left(16x^{2}\right)^{2}}$$

Do the arithmetic.

$$\frac{16x^{2}x^{0}-\left(x^{1}+22\right)\times 32x^{1}}{\left(16x^{2}\right)^{2}}$$

Expand using distributive property.

$$\frac{16x^{2}x^{0}-\left(x^{1}\times 32x^{1}+22\times 32x^{1}\right)}{\left(16x^{2}\right)^{2}}$$

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

$$\frac{16x^{2}-\left(32x^{1+1}+22\times 32x^{1}\right)}{\left(16x^{2}\right)^{2}}$$

Do the arithmetic.

$$\frac{16x^{2}-\left(32x^{2}+704x^{1}\right)}{\left(16x^{2}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{16x^{2}-32x^{2}-704x^{1}}{\left(16x^{2}\right)^{2}}$$

Combine like terms.

$$\frac{\left(16-32\right)x^{2}-704x^{1}}{\left(16x^{2}\right)^{2}}$$

Subtract 32 from 16.

$$\frac{-16x^{2}-704x^{1}}{\left(16x^{2}\right)^{2}}$$

Factor out 16x.

$$\frac{16x\left(-x^{1}-44x^{0}\right)}{\left(16x^{2}\right)^{2}}$$

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

$$\frac{16x\left(-x^{1}-44x^{0}\right)}{16^{2}\left(x^{2}\right)^{2}}$$

Raise 16 to the power 2.

$$\frac{16x\left(-x^{1}-44x^{0}\right)}{256\left(x^{2}\right)^{2}}$$

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

$$\frac{16x\left(-x^{1}-44x^{0}\right)}{256x^{2\times 2}}$$

Multiply 2 times 2.

$$\frac{16x\left(-x^{1}-44x^{0}\right)}{256x^{4}}$$

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

$$\frac{16\left(-x^{1}-44x^{0}\right)}{256x^{4-1}}$$

Subtract 1 from 4.

$$\frac{16\left(-x^{1}-44x^{0}\right)}{256x^{3}}$$

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

$$\frac{16\left(-x-44x^{0}\right)}{256x^{3}}$$

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

$$\frac{16\left(-x-44\times 1\right)}{256x^{3}}$$

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

$$\frac{16\left(-x-44\right)}{256x^{3}}$$

Differentiate w.r.t. x

$$\frac{x+66}{8x^{4}}$$