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

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

Evaluate

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

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).

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

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}.

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

Simplify.

$$\left(x^{2}+3x^{1}-2\right)^{2}\left(6x^{1}+9x^{0}\right)$$

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

$$\left(x^{2}+3x-2\right)^{2}\left(6x+9x^{0}\right)$$

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

$$\left(x^{2}+3x-2\right)^{2}\left(6x+9\times 1\right)$$

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

$$\left(x^{2}+3x-2\right)^{2}\left(6x+9\right)$$

Differentiate w.r.t. x

$$30x^{4}+180x^{3}+252x^{2}-54x-84$$