Solução - Derivada

A derivada de um valor constante é sempre zero.

$$\frac{d}{dx}\left[4\right]\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \frac{d}{dx}\left[{\left(2x^2+5\right)}^3\right]\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \frac{d}{dx}\left[{\left(2x^2+5\right)}^3\right]\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]$$

Calculando a derivada de uma função potência.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \frac{d}{dx}\left[{\left(2x^2+5\right)}^3\right]\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]$$

Aplicando a regra do produto das derivadas.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \frac{d}{dx}\left[4x\right]=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[4\right]\times x+4\times \frac{d}{dx}\left[x\right]\right)$$

A derivada de um valor constante é sempre zero.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[4\right]\times x+4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0x+4\times \frac{d}{dx}\left[x\right]\right)$$

Multiplicar um número por zero sempre resulta em zero.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0x+4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0+4\times \frac{d}{dx}\left[x\right]\right)$$

Adicionando zero a um número, o que não muda o seu valor.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\left(0+4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times \frac{d}{dx}\left[x\right]\right)$$

A derivada de uma variável em relação a ela mesma é sempre igual a um.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times \frac{d}{dx}\left[x\right]\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times 1\right)$$

Multiplicar um número por um, o que não muda o seu valor.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times \left(4\times 1\right)=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

A derivada de um valor constante é sempre zero.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(\frac{d}{dx}\left[3\right]\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Aplicando a regra da soma das derivadas.

$$0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \frac{d}{dx}[2x^2+5]\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0{\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Multiplicar um número por zero sempre resulta em zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0\times \ln \left(2x^2+5\right)+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Aplicando a regra do produto das derivadas.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\frac{d}{dx}\left[2x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(\frac{d}{dx}\left[2\right]\times x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

A derivada de um valor constante é sempre zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(\frac{d}{dx}\left[2\right]\times x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Multiplicar um número por zero sempre resulta em zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0x^2+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Adicionando zero a um número, o que não muda o seu valor.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(0+2\times \frac{d}{dx}\left[x^2\right]\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Calculando a derivada de x elevado à potência de n.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^{2-1}\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Subtraindo um de um número.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^{2-1}\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^1\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

A multiplicação pode ser agrupada de maneira diferente, mas o resultado permanece o mesmo.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(2\times \left(2x^1\right)+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(2\times 2\right)\times x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Multiplicando dois números inteiros juntos.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(\left(2\times 2\right)\times x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(4x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

A derivada de um valor constante é sempre zero.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(4x^1+\frac{d}{dx}\left[5\right]\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\left(0+\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4{\left(2x^2+5\right)}^3\times 4$$

Adicionando zero a um número, o que não muda o seu valor.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\times \left(0+\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4$$

Simplificando as expressões aritméticas.

$$0\times {\left(2x^2+5\right)}^3\times \left(4x\right)+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4$$

Adicionando zero a um número, o que não muda o seu valor.

$$0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1+0\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4$$

Simplificando as expressões aritméticas.

$$0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+4\times {\left(2x^2+5\right)}^3\times 4=0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Adicionando zero a um número, o que não muda o seu valor.

$$0+4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificando as expressões aritméticas.

$$4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x^1\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificando as expressões aritméticas.

$$4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{3}{2x^2+5}\times \left(4x\right)\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{12x}{2x^2+5}\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificando as expressões aritméticas.

$$4\times \left({\left(2x^2+5\right)}^3\times \left(\frac{12x}{2x^2+5}\right)\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=4\times \left(12x{\left(2x^2+5\right)}^2\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3$$

Simplificando as expressões aritméticas.

$$4\times \left(12x{\left(2x^2+5\right)}^2\right)\times \left(4x\right)+16{\left(2x^2+5\right)}^3=192x^2{\left(2x^2+5\right)}^2+16{\left(2x^2+5\right)}^3$$