Solve (9c^5)^frac{1{2}}

$$( 9 c ^ { 5 } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$3c^{\frac{5}{2}}$$

Solution Steps

Expand \left(9c^{5}\right)^{\frac{1}{2}}.

$$9^{\frac{1}{2}}\left(c^{5}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 5 and \frac{1}{2} to get \frac{5}{2}.

$$9^{\frac{1}{2}}c^{\frac{5}{2}}$$

Calculate 9 to the power of \frac{1}{2} and get 3.

$$3c^{\frac{5}{2}}$$

Differentiate w.r.t. c

$$\frac{15c^{\frac{3}{2}}}{2}$$

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).

$$\frac{1}{2}\times \left(9c^{5}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}c}(9c^{5})$$

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}.

$$\frac{1}{2}\times \left(9c^{5}\right)^{-\frac{1}{2}}\times 5\times 9c^{5-1}$$

Simplify.

$$\frac{45}{2}c^{4}\times \left(9c^{5}\right)^{-\frac{1}{2}}$$