Solve (9r^4)^0,5

$$( 9 r ^ { 4 } ) ^ { 0,5 }$$

Evaluate

$$3r^{2}$$

Solution Steps

Expand \left(9r^{4}\right)^{0,5}.

$$9^{0,5}\left(r^{4}\right)^{0,5}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 0,5 to get 2.

$$9^{0,5}r^{2}$$

Calculate 9 to the power of 0,5 and get 3.

$$3r^{2}$$

Differentiate w.r.t. r

$$6r$$

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).

$$0,5\times \left(9r^{4}\right)^{0,5-1}\frac{\mathrm{d}}{\mathrm{d}r}(9r^{4})$$

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}.

$$0,5\times \left(9r^{4}\right)^{-0,5}\times 4\times 9r^{4-1}$$

Simplify.

$$18r^{3}\times \left(9r^{4}\right)^{-0,5}$$