Solve (121x^4)^frac{1{2}}=

$$( 121 x ^ { 4 } ) ^ { \frac { 1 } { 2 } } =$$

Evaluate

$$11x^{2}$$

Solution Steps

Expand \left(121x^{4}\right)^{\frac{1}{2}}.

$$121^{\frac{1}{2}}\left(x^{4}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{1}{2} to get 2.

$$121^{\frac{1}{2}}x^{2}$$

Calculate 121 to the power of \frac{1}{2} and get 11.

$$11x^{2}$$

Differentiate w.r.t. x

$$22x$$

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).

$$\frac{1}{2}\times \left(121x^{4}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}x}(121x^{4})$$

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}.

$$\frac{1}{2}\times \left(121x^{4}\right)^{-\frac{1}{2}}\times 4\times 121x^{4-1}$$

Simplify.

$$242x^{3}\times \left(121x^{4}\right)^{-\frac{1}{2}}$$