Solve 1/(-6x^7)(5)(2x^8/5)

$$\frac { 1 } { ( - 6 x ^ { 7 } ) ( 5 ) ( 2 x ^ { 8 } / 5 ) }$$

Evaluate

$$-\frac{1}{12x^{15}}$$

Solution Steps

Multiply -6 and 5 to get -30.

$$\frac{1}{-30x^{7}\times \frac{2x^{8}}{5}}$$

Cancel out 5, the greatest common factor in 30 and 5.

$$\frac{1}{-6\times 2x^{8}x^{7}}$$

To multiply powers of the same base, add their exponents. Add 8 and 7 to get 15.

$$\frac{1}{-6\times 2x^{15}}$$

Multiply -6 and 2 to get -12.

$$\frac{1}{-12x^{15}}$$

Differentiate w.r.t. x

$$\frac{5}{4x^{16}}$$

Steps Using Chain Rule

Multiply -6 and 5 to get -30.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{-30x^{7}\times \frac{2x^{8}}{5}})$$

Cancel out 5, the greatest common factor in 30 and 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{-6\times 2x^{8}x^{7}})$$

To multiply powers of the same base, add their exponents. Add 8 and 7 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{-6\times 2x^{15}})$$

Multiply -6 and 2 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{-12x^{15}})$$

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).

$$-\left(-12x^{15}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(-12x^{15})$$

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}.

$$-\left(-12x^{15}\right)^{-2}\times 15\left(-12\right)x^{15-1}$$

Simplify.

$$180x^{14}\left(-12x^{15}\right)^{-2}$$