Solve [(x^3-1)^2+4(x^3-1)]^prime

$$[ ( x ^ { 3 } - 1 ) ^ { 2 } + 4 ( x ^ { 3 } - 1 ) ] ^ { \prime }$$

Evaluate

$$6x^{2}\left(x^{3}+1\right)$$

Steps Using Derivative Rule for Sum

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{3}-1\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}-2x^{3}+1+4\left(x^{3}-1\right))$$

Use the distributive property to multiply 4 by x^{3}-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}-2x^{3}+1+4x^{3}-4)$$

Combine -2x^{3} and 4x^{3} to get 2x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+2x^{3}+1-4)$$

Subtract 4 from 1 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6}+2x^{3}-3)$$

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}.

$$6x^{6-1}+3\times 2x^{3-1}$$

Subtract 1 from 6.

$$6x^{5}+3\times 2x^{3-1}$$

Multiply 3 times 2.

$$6x^{5}+6x^{3-1}$$

Subtract 1 from 3.

$$6x^{5}+6x^{2}$$

Differentiate w.r.t. x

$$30x^{4}+12x$$