Solve 3s^5t/3s^5t^7=

$$\frac { 3 s ^ { 5 } t } { 3 s ^ { 5 } t ^ { 7 } } =$$

Evaluate

$$\frac{1}{t^{6}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{3^{1}s^{5}t^{1}}{3^{1}s^{5}t^{7}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$3^{1-1}s^{5-5}t^{1-7}$$

Subtract 1 from 1.

$$3^{0}s^{5-5}t^{1-7}$$

For any number a except 0, a^{0}=1.

$$s^{5-5}t^{1-7}$$

Subtract 5 from 5.

$$s^{0}t^{1-7}$$

For any number a except 0, a^{0}=1.

$$t^{1-7}$$

Subtract 7 from 1.

$$s^{0}t^{-6}$$

For any term t except 0, t^{0}=1.

$$1t^{-6}$$

For any term t, t\times 1=t and 1t=t.

$$t^{-6}$$

Differentiate w.r.t. t

$$-\frac{6}{t^{7}}$$

Steps Using Chain Rule

Cancel out 3ts^{5} in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

$$-6t^{5}\left(t^{6}\right)^{-2}$$