Solve 4(q^3)^2/12q^10

$$\frac { 4 ( q ^ { 3 } ) ^ { 2 } } { 12 q ^ { 10 } }$$

Evaluate

$$\frac{1}{3q^{4}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{4q^{6}}{12q^{10}}$$

Cancel out 4q^{6} in both numerator and denominator.

$$\frac{1}{3q^{4}}$$

Differentiate w.r.t. q

$$-\frac{4}{3q^{5}}$$

Steps Using Chain Rule

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}q}(\frac{4q^{6}}{12q^{10}})$$

Cancel out 4q^{6} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}q}(\frac{1}{3q^{4}})$$

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).

$$-\left(3q^{4}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}q}(3q^{4})$$

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}.

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

Simplify.

$$-12q^{3}\times \left(3q^{4}\right)^{-2}$$