Solve 5p^2-6p/5p-6

$$\frac { 5 p ^ { 2 } - 6 p } { 5 p - 6 }$$

Evaluate

$$p$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{p\left(5p-6\right)}{5p-6}$$

Cancel out 5p-6 in both numerator and denominator.

$$p$$

Differentiate w.r.t. p

$$1$$

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.

$$\frac{\left(5p^{1}-6\right)\frac{\mathrm{d}}{\mathrm{d}p}(5p^{2}-6p^{1})-\left(5p^{2}-6p^{1}\right)\frac{\mathrm{d}}{\mathrm{d}p}(5p^{1}-6)}{\left(5p^{1}-6\right)^{2}}$$

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}.

$$\frac{\left(5p^{1}-6\right)\left(2\times 5p^{2-1}-6p^{1-1}\right)-\left(5p^{2}-6p^{1}\right)\times 5p^{1-1}}{\left(5p^{1}-6\right)^{2}}$$

Simplify.

$$\frac{\left(5p^{1}-6\right)\left(10p^{1}-6p^{0}\right)-\left(5p^{2}-6p^{1}\right)\times 5p^{0}}{\left(5p^{1}-6\right)^{2}}$$

Multiply 5p^{1}-6 times 10p^{1}-6p^{0}.

$$\frac{5p^{1}\times 10p^{1}+5p^{1}\left(-6\right)p^{0}-6\times 10p^{1}-6\left(-6\right)p^{0}-\left(5p^{2}-6p^{1}\right)\times 5p^{0}}{\left(5p^{1}-6\right)^{2}}$$

Multiply 5p^{2}-6p^{1} times 5p^{0}.

$$\frac{5p^{1}\times 10p^{1}+5p^{1}\left(-6\right)p^{0}-6\times 10p^{1}-6\left(-6\right)p^{0}-\left(5p^{2}\times 5p^{0}-6p^{1}\times 5p^{0}\right)}{\left(5p^{1}-6\right)^{2}}$$

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

$$\frac{5\times 10p^{1+1}+5\left(-6\right)p^{1}-6\times 10p^{1}-6\left(-6\right)p^{0}-\left(5\times 5p^{2}-6\times 5p^{1}\right)}{\left(5p^{1}-6\right)^{2}}$$

Simplify.

$$\frac{50p^{2}-30p^{1}-60p^{1}+36p^{0}-\left(25p^{2}-30p^{1}\right)}{\left(5p^{1}-6\right)^{2}}$$

Combine like terms.

$$\frac{25p^{2}-60p^{1}+36p^{0}}{\left(5p^{1}-6\right)^{2}}$$

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

$$\frac{25p^{2}-60p+36p^{0}}{\left(5p-6\right)^{2}}$$

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

$$\frac{25p^{2}-60p+36\times 1}{\left(5p-6\right)^{2}}$$

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

$$\frac{25p^{2}-60p+36}{\left(5p-6\right)^{2}}$$