Solve 10x^{7}:(x^{3}-6x^3)+4x^3*5x-(3x^2*x^2)

$$10 x ^ { 7 } : ( x ^ { 3 } - 6 x ^ { 3 } ) + 4 x ^ { 3 } \cdot 5 x - ( 3 x ^ { 2 } \cdot x ^ { 2 } )$$

Evaluate

$$15x^{4}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{10x^{7}}{x^{3}-6x^{3}}+4x^{4}\times 5-3x^{2}x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{10x^{7}}{x^{3}-6x^{3}}+4x^{4}\times 5-3x^{4}$$

Combine x^{3} and -6x^{3} to get -5x^{3}.

$$\frac{10x^{7}}{-5x^{3}}+4x^{4}\times 5-3x^{4}$$

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

$$\frac{2x^{4}}{-1}+4x^{4}\times 5-3x^{4}$$

Anything divided by -1 gives its opposite.

$$-2x^{4}+4x^{4}\times 5-3x^{4}$$

Multiply 4 and 5 to get 20.

$$-2x^{4}+20x^{4}-3x^{4}$$

Combine -2x^{4} and 20x^{4} to get 18x^{4}.

$$18x^{4}-3x^{4}$$

Combine 18x^{4} and -3x^{4} to get 15x^{4}.

$$15x^{4}$$

Differentiate w.r.t. x

$$60x^{3}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{7}}{x^{3}-6x^{3}}+4x^{4}\times 5-3x^{2}x^{2})$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{7}}{x^{3}-6x^{3}}+4x^{4}\times 5-3x^{4})$$

Combine x^{3} and -6x^{3} to get -5x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{10x^{7}}{-5x^{3}}+4x^{4}\times 5-3x^{4})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{4}}{-1}+4x^{4}\times 5-3x^{4})$$

Anything divided by -1 gives its opposite.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-2x^{4}+4x^{4}\times 5-3x^{4})$$

Multiply 4 and 5 to get 20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-2x^{4}+20x^{4}-3x^{4})$$

Combine -2x^{4} and 20x^{4} to get 18x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(18x^{4}-3x^{4})$$

Combine 18x^{4} and -3x^{4} to get 15x^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(15x^{4})$$

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

$$4\times 15x^{4-1}$$

Multiply 4 times 15.

$$60x^{4-1}$$

Subtract 1 from 4.

$$60x^{3}$$