Solve 4x^3+4x^2+x/4x+1*4x+1/4x^2+4x+1

$$\frac { 4 x ^ { 3 } + 4 x ^ { 2 } + x } { 4 x + 1 } \cdot \frac { 4 x + 1 } { 4 x ^ { 2 } + 4 x + 1 }$$

Evaluate

$$x$$

Short Solution Steps

Multiply \frac{4x^{3}+4x^{2}+x}{4x+1} times \frac{4x+1}{4x^{2}+4x+1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4x^{3}+4x^{2}+x\right)\left(4x+1\right)}{\left(4x+1\right)\left(4x^{2}+4x+1\right)}$$

Cancel out 4x+1 in both numerator and denominator.

$$\frac{4x^{3}+4x^{2}+x}{4x^{2}+4x+1}$$

Factor the expressions that are not already factored.

$$\frac{x\left(2x+1\right)^{2}}{\left(2x+1\right)^{2}}$$

Cancel out \left(2x+1\right)^{2} in both numerator and denominator.

$$x$$

Differentiate w.r.t. x

$$1$$

Steps Using Definition of a Derivative

Multiply \frac{4x^{3}+4x^{2}+x}{4x+1} times \frac{4x+1}{4x^{2}+4x+1} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(4x^{3}+4x^{2}+x\right)\left(4x+1\right)}{\left(4x+1\right)\left(4x^{2}+4x+1\right)})$$

Cancel out 4x+1 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{3}+4x^{2}+x}{4x^{2}+4x+1})$$

Factor the expressions that are not already factored in \frac{4x^{3}+4x^{2}+x}{4x^{2}+4x+1}.

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

Cancel out \left(2x+1\right)^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x)$$

The derivative of ax^{n} is nax^{n-1}.

$$x^{1-1}$$

Subtract 1 from 1.

$$x^{0}$$

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

$$1$$