Solve (4a^2)^frac{-1{2}}

$$( 4 a ^ { 2 } ) ^ { \frac { - 1 } { 2 } }$$

Evaluate

$$\frac{1}{2|a|}$$

Solution Steps

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$\left(4a^{2}\right)^{-\frac{1}{2}}$$

Expand \left(4a^{2}\right)^{-\frac{1}{2}}.

$$4^{-\frac{1}{2}}\left(a^{2}\right)^{-\frac{1}{2}}$$

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

$$4^{-\frac{1}{2}}a^{-1}$$

Calculate 4 to the power of -\frac{1}{2} and get \frac{1}{2}.

$$\frac{1}{2}a^{-1}$$

Differentiate w.r.t. a

$$-\frac{|a|}{2a^{3}}$$

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).

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

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}.

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

Simplify.

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

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

$$-4a\times \left(4a^{2}\right)^{-\frac{3}{2}}$$