Solve f(t)=(3t^2-2t+1)^3/2

$$f ( t ) = ( 3 t ^ { 2 } - 2 t + 1 ) ^ { 3 / 2 }$$

Differentiate w.r.t. t

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

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).

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

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}.

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

Simplify.

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

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

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

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

$$\sqrt{3t^{2}-2t+1}\left(9t-3\right)$$

Evaluate

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