Penyelesaian - Derivative

Computing the derivative of a power function.

$$\frac{d}{dx}\left[{\left(x^3+1\right)}^7\right]={\left(x^3+1\right)}^7\left(\frac{d}{dx}\left[7\right]\times \ln \left(x^3+1\right)+\frac{7}{x^3+1}\times \frac{d}{dx}[x^3+1]\right)$$

The derivative of a constant value is always zero.

$${\left(x^3+1\right)}^7\left(\frac{d}{dx}\left[7\right]\times \ln \left(x^3+1\right)+\frac{7}{x^3+1}\times \frac{d}{dx}[x^3+1]\right)={\left(x^3+1\right)}^7\left(0\times \ln \left(x^3+1\right)+\frac{7}{x^3+1}\times \frac{d}{dx}[x^3+1]\right)$$

Applying the sum rule of derivatives.

$${\left(x^3+1\right)}^7\left(0\times \ln \left(x^3+1\right)+\frac{7}{x^3+1}\times \frac{d}{dx}[x^3+1]\right)={\left(x^3+1\right)}^7\left(0\times \ln \left(x^3+1\right)+\frac{7}{x^3+1}\times \left(\frac{d}{dx}\left[x^3\right]+\frac{d}{dx}\left[1\right]\right)\right)$$

Multiplying a number by zero always results in zero.

$${\left(x^3+1\right)}^7\times \left(0\times \ln \left(x^3+1\right)+\frac{7}{x^3+1}\times \left(\frac{d}{dx}\left[x^3\right]+\frac{d}{dx}\left[1\right]\right)\right)={\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(\frac{d}{dx}\left[x^3\right]+\frac{d}{dx}\left[1\right]\right)\right)$$

Computing the derivative of x raised to the power of n.

$${\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(\frac{d}{dx}\left[x^3\right]+\frac{d}{dx}\left[1\right]\right)\right)={\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(3x^{3-1}+\frac{d}{dx}\left[1\right]\right)\right)$$

Subtracting one from a number.

$${\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(3x^{3-1}+\frac{d}{dx}\left[1\right]\right)\right)={\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(3x^2+\frac{d}{dx}\left[1\right]\right)\right)$$

The derivative of a constant value is always zero.

$${\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(3x^2+\frac{d}{dx}\left[1\right]\right)\right)={\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(3x^2+0\right)\right)$$

Adding zero to a number, which does not change its value.

$${\left(x^3+1\right)}^7\times \left(0+\frac{7}{x^3+1}\times \left(3x^2+0\right)\right)={\left(x^3+1\right)}^7\times \left(\frac{7}{x^3+1}\times \left(3x^2+0\right)\right)$$

Adding zero to a number, which does not change its value.

$${\left(x^3+1\right)}^7\times \left(\frac{7}{x^3+1}\times \left(3x^2+0\right)\right)={\left(x^3+1\right)}^7\times \left(\frac{7}{x^3+1}\times \left(3x^2\right)\right)$$

Simplifying the arithmetic expressions.

$${\left(x^3+1\right)}^7\times \left(\frac{7}{x^3+1}\times \left(3x^2\right)\right)={\left(x^3+1\right)}^7\times \left(\frac{21x^2}{x^3+1}\right)$$

Simplifying the arithmetic expressions.

$${\left(x^3+1\right)}^7\times \left(\frac{21x^2}{x^3+1}\right)=21x^2{\left(x^3+1\right)}^6$$