Solve 10^-63^-7625*x^-4div5^-3*{6}^-5*{x}^-8

$${ 10 }^{ -6 } { 3 }^{ -7 } 625 \times { x }^{ -4 } \div { 5 }^{ -3 } \times { 6 }^{ -5 } \times { x }^{ -8 }$$

Evaluate

$$\frac{5}{1088391168x^{12}}$$

Solution Steps

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\frac{\frac{1}{1000000}\times 3^{-7}\times 625x^{-4}}{5^{-3}}\times 6^{-5}x^{-8}$$

Calculate 3 to the power of -7 and get \frac{1}{2187}.

$$\frac{\frac{1}{1000000}\times \frac{1}{2187}\times 625x^{-4}}{5^{-3}}\times 6^{-5}x^{-8}$$

Multiply \frac{1}{1000000} and \frac{1}{2187} to get \frac{1}{2187000000}.

$$\frac{\frac{1}{2187000000}\times 625x^{-4}}{5^{-3}}\times 6^{-5}x^{-8}$$

Multiply \frac{1}{2187000000} and 625 to get \frac{1}{3499200}.

$$\frac{\frac{1}{3499200}x^{-4}}{5^{-3}}\times 6^{-5}x^{-8}$$

Calculate 5 to the power of -3 and get \frac{1}{125}.

$$\frac{\frac{1}{3499200}x^{-4}}{\frac{1}{125}}\times 6^{-5}x^{-8}$$

Divide \frac{1}{3499200}x^{-4} by \frac{1}{125} by multiplying \frac{1}{3499200}x^{-4} by the reciprocal of \frac{1}{125}.

$$\frac{1}{3499200}x^{-4}\times 125\times 6^{-5}x^{-8}$$

Multiply \frac{1}{3499200} and 125 to get \frac{5}{139968}.

$$\frac{5}{139968}x^{-4}\times 6^{-5}x^{-8}$$

Calculate 6 to the power of -5 and get \frac{1}{7776}.

$$\frac{5}{139968}x^{-4}\times \frac{1}{7776}x^{-8}$$

Multiply \frac{5}{139968} and \frac{1}{7776} to get \frac{5}{1088391168}.

$$\frac{5}{1088391168}x^{-4}x^{-8}$$

To multiply powers of the same base, add their exponents. Add -4 and -8 to get -12.

$$\frac{5}{1088391168}x^{-12}$$

Differentiate w.r.t. x

$$-\frac{5}{90699264x^{13}}$$

Steps Using Definition of a Derivative

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{1000000}\times 3^{-7}\times 625x^{-4}}{5^{-3}}\times 6^{-5}x^{-8})$$

Calculate 3 to the power of -7 and get \frac{1}{2187}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{1000000}\times \frac{1}{2187}\times 625x^{-4}}{5^{-3}}\times 6^{-5}x^{-8})$$

Multiply \frac{1}{1000000} and \frac{1}{2187} to get \frac{1}{2187000000}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{2187000000}\times 625x^{-4}}{5^{-3}}\times 6^{-5}x^{-8})$$

Multiply \frac{1}{2187000000} and 625 to get \frac{1}{3499200}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{3499200}x^{-4}}{5^{-3}}\times 6^{-5}x^{-8})$$

Calculate 5 to the power of -3 and get \frac{1}{125}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{3499200}x^{-4}}{\frac{1}{125}}\times 6^{-5}x^{-8})$$

Divide \frac{1}{3499200}x^{-4} by \frac{1}{125} by multiplying \frac{1}{3499200}x^{-4} by the reciprocal of \frac{1}{125}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3499200}x^{-4}\times 125\times 6^{-5}x^{-8})$$

Multiply \frac{1}{3499200} and 125 to get \frac{5}{139968}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{139968}x^{-4}\times 6^{-5}x^{-8})$$

Calculate 6 to the power of -5 and get \frac{1}{7776}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{139968}x^{-4}\times \frac{1}{7776}x^{-8})$$

Multiply \frac{5}{139968} and \frac{1}{7776} to get \frac{5}{1088391168}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{1088391168}x^{-4}x^{-8})$$

To multiply powers of the same base, add their exponents. Add -4 and -8 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{1088391168}x^{-12})$$

The derivative of ax^{n} is nax^{n-1}.

$$-12\times \frac{5}{1088391168}x^{-12-1}$$

Multiply -12 times \frac{5}{1088391168}.

$$-\frac{5}{90699264}x^{-12-1}$$

Subtract 1 from -12.

$$-\frac{5}{90699264}x^{-13}$$