Solve d(x-2x^2)^3/dx

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

Evaluate

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

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

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}.

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

Simplify.

$$\left(x^{1}-2x^{2}\right)^{2}\left(3x^{0}-12x^{1}\right)$$

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

$$\left(x-2x^{2}\right)^{2}\left(3x^{0}-12x\right)$$

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

$$\left(x-2x^{2}\right)^{2}\left(3\times 1-12x\right)$$

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

$$\left(x-2x^{2}\right)^{2}\left(3-12x\right)$$

Differentiate w.r.t. x

$$6x\left(2x-1\right)\left(-20x^{2}+10x-1\right)$$