Solve (x+1)(x+1)^1/2=

$$( x + 1 ) ( x + 1 ) ^ { 1 / 2 } =$$

Differentiate w.r.t. x

$$\frac{3\sqrt{x+1}}{2}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add 1 and \frac{1}{2} to get \frac{3}{2}.

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

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).

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

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}.

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

Simplify.

$$\frac{3}{2}x^{0}\sqrt{x^{1}+1}$$

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

$$\frac{3}{2}x^{0}\sqrt{x+1}$$

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

$$\frac{3}{2}\times 1\sqrt{x+1}$$

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

$$\frac{3}{2}\sqrt{x+1}$$

Evaluate

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