Solve h(x)=(5x-3)^3/2

$$h ( x ) = ( 5 x - 3 ) ^ { 3 / 2 }$$

Differentiate w.r.t. x

$$\frac{15\sqrt{5x-3}}{2}$$

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).

$$\frac{3}{2}\left(5x^{1}-3\right)^{\frac{3}{2}-1}\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}-3)$$

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}.

$$\frac{3}{2}\sqrt{5x^{1}-3}\times 5x^{1-1}$$

Simplify.

$$\frac{15}{2}x^{0}\sqrt{5x^{1}-3}$$

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

$$\frac{15}{2}x^{0}\sqrt{5x-3}$$

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

$$\frac{15}{2}\times 1\sqrt{5x-3}$$

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

$$\frac{15}{2}\sqrt{5x-3}$$

Evaluate

$$\left(5x-3\right)^{\frac{3}{2}}$$