Solve 1/2a*1/3a

$$\frac { 1 } { 2 a } \times \frac { 1 } { 3 a }$$

Evaluate

$$\frac{1}{6a^{2}}$$

Solution Steps

Multiply \frac{1}{2a} times \frac{1}{3a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{2a\times 3a}$$

Multiply a and a to get a^{2}.

$$\frac{1}{2a^{2}\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{1}{6a^{2}}$$

Differentiate w.r.t. a

$$-\frac{1}{3a^{3}}$$

Steps Using Chain Rule

Multiply \frac{1}{2a} times \frac{1}{3a} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{2a\times 3a})$$

Multiply a and a to get a^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{2a^{2}\times 3})$$

Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{6a^{2}})$$

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).

$$-\left(6a^{2}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(6a^{2})$$

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}.

$$-\left(6a^{2}\right)^{-2}\times 2\times 6a^{2-1}$$

Simplify.

$$-12a^{1}\times \left(6a^{2}\right)^{-2}$$

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

$$-12a\times \left(6a^{2}\right)^{-2}$$