Solve =(3x^2-4x+3)^3/2

$$= ( 3 x ^ { 2 } - 4 x + 3 ) ^ { 3 / 2 }$$

Differentiate w.r.t. x

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

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).

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

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}.

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

Simplify.

$$\sqrt{3x^{2}-4x^{1}+3}\left(9x^{1}-6x^{0}\right)$$

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

$$\sqrt{3x^{2}-4x+3}\left(9x-6x^{0}\right)$$

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

$$\sqrt{3x^{2}-4x+3}\left(9x-6\right)$$

Evaluate

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