Solve 40j/-70-70j

$$\frac { 40 j } { - 70 - 70 j }$$

Evaluate

$$-\frac{4j}{7\left(j+1\right)}$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{40j}{70\left(-j-1\right)}$$

Cancel out 10 in both numerator and denominator.

$$\frac{4j}{7\left(-j-1\right)}$$

Expand the expression.

$$\frac{4j}{-7j-7}$$

Differentiate w.r.t. j

$$-\frac{4}{7\left(j+1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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.

$$\frac{\left(-70j^{1}-70\right)\frac{\mathrm{d}}{\mathrm{d}j}(40j^{1})-40j^{1}\frac{\mathrm{d}}{\mathrm{d}j}(-70j^{1}-70)}{\left(-70j^{1}-70\right)^{2}}$$

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}.

$$\frac{\left(-70j^{1}-70\right)\times 40j^{1-1}-40j^{1}\left(-70\right)j^{1-1}}{\left(-70j^{1}-70\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(-70j^{1}-70\right)\times 40j^{0}-40j^{1}\left(-70\right)j^{0}}{\left(-70j^{1}-70\right)^{2}}$$

Expand using distributive property.

$$\frac{-70j^{1}\times 40j^{0}-70\times 40j^{0}-40j^{1}\left(-70\right)j^{0}}{\left(-70j^{1}-70\right)^{2}}$$

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

$$\frac{-70\times 40j^{1}-70\times 40j^{0}-40\left(-70\right)j^{1}}{\left(-70j^{1}-70\right)^{2}}$$

Do the arithmetic.

$$\frac{-2800j^{1}-2800j^{0}-\left(-2800j^{1}\right)}{\left(-70j^{1}-70\right)^{2}}$$

Combine like terms.

$$\frac{\left(-2800-\left(-2800\right)\right)j^{1}-2800j^{0}}{\left(-70j^{1}-70\right)^{2}}$$

Subtract -2800 from -2800.

$$\frac{-2800j^{0}}{\left(-70j^{1}-70\right)^{2}}$$

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

$$\frac{-2800j^{0}}{\left(-70j-70\right)^{2}}$$

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

$$\frac{-2800}{\left(-70j-70\right)^{2}}$$