Solve frac{x^{3}*y^{2}*y^5*x^-1*x^4}{y^7}

$$\frac { x ^ { 3 } \cdot y ^ { 2 } \cdot y ^ { 5 } \cdot x ^ { - 1 } \cdot x ^ { 4 } } { y ^ { 7 } }$$

Evaluate

$$x^{6}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and -1 to get 2.

$$\frac{x^{2}y^{2}y^{5}x^{4}}{y^{7}}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{x^{6}y^{2}y^{5}}{y^{7}}$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{x^{6}y^{7}}{y^{7}}$$

Cancel out y^{7} in both numerator and denominator.

$$x^{6}$$

Differentiate w.r.t. x

$$6x^{5}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and -1 to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}y^{2}y^{5}x^{4}}{y^{7}})$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{6}y^{2}y^{5}}{y^{7}})$$

To multiply powers of the same base, add their exponents. Add 2 and 5 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{6}y^{7}}{y^{7}})$$

Cancel out y^{7} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{6})$$

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

$$6x^{6-1}$$

Subtract 1 from 6.

$$6x^{5}$$