Solve (1/x-7)(1/x-6)

$$( \frac { 1 } { x - 7 } ) ( \frac { 1 } { x - 6 } )$$

Evaluate

$$\frac{1}{\left(x-7\right)\left(x-6\right)}$$

Solution Steps

Multiply \frac{1}{x-7} times \frac{1}{x-6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{\left(x-7\right)\left(x-6\right)}$$

Apply the distributive property by multiplying each term of x-7 by each term of x-6.

$$\frac{1}{x^{2}-6x-7x+42}$$

Combine -6x and -7x to get -13x.

$$\frac{1}{x^{2}-13x+42}$$

Differentiate w.r.t. x

$$\frac{13-2x}{\left(\left(x-7\right)\left(x-6\right)\right)^{2}}$$

Steps Using Chain Rule

Multiply \frac{1}{x-7} times \frac{1}{x-6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\left(x-7\right)\left(x-6\right)})$$

Apply the distributive property by multiplying each term of x-7 by each term of x-6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{2}-6x-7x+42})$$

Combine -6x and -7x to get -13x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{2}-13x+42})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(x^{2}-13x^{1}+42\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-13x^{1}+42)$$

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}.

$$-\left(x^{2}-13x^{1}+42\right)^{-2}\left(2x^{2-1}-13x^{1-1}\right)$$

Simplify.

$$\left(x^{2}-13x^{1}+42\right)^{-2}\left(-2x^{1}+13x^{0}\right)$$

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

$$\left(x^{2}-13x+42\right)^{-2}\left(-2x+13x^{0}\right)$$

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

$$\left(x^{2}-13x+42\right)^{-2}\left(-2x+13\times 1\right)$$

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

$$\left(x^{2}-13x+42\right)^{-2}\left(-2x+13\right)$$