Solve 3^5alpha10^5*25/5^7*6^5

$$\frac { 3 ^ { 5 } \alpha 10 ^ { 5 } \times 25 } { 5 ^ { 7 } \times 6 ^ { 5 } }$$

Evaluate

$$\alpha$$

Solution Steps

Calculate 3 to the power of 5 and get 243.

$$\frac{243\alpha \times 10^{5}\times 25}{5^{7}\times 6^{5}}$$

Calculate 10 to the power of 5 and get 100000.

$$\frac{243\alpha \times 100000\times 25}{5^{7}\times 6^{5}}$$

Multiply 243 and 100000 to get 24300000.

$$\frac{24300000\alpha \times 25}{5^{7}\times 6^{5}}$$

Multiply 24300000 and 25 to get 607500000.

$$\frac{607500000\alpha }{5^{7}\times 6^{5}}$$

Calculate 5 to the power of 7 and get 78125.

$$\frac{607500000\alpha }{78125\times 6^{5}}$$

Calculate 6 to the power of 5 and get 7776.

$$\frac{607500000\alpha }{78125\times 7776}$$

Multiply 78125 and 7776 to get 607500000.

$$\frac{607500000\alpha }{607500000}$$

Cancel out 607500000 and 607500000.

$$\alpha$$

Differentiate w.r.t. α

$$1$$

Steps Using Definition of a Derivative

Calculate 3 to the power of 5 and get 243.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{243\alpha \times 10^{5}\times 25}{5^{7}\times 6^{5}})$$

Calculate 10 to the power of 5 and get 100000.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{243\alpha \times 100000\times 25}{5^{7}\times 6^{5}})$$

Multiply 243 and 100000 to get 24300000.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{24300000\alpha \times 25}{5^{7}\times 6^{5}})$$

Multiply 24300000 and 25 to get 607500000.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{607500000\alpha }{5^{7}\times 6^{5}})$$

Calculate 5 to the power of 7 and get 78125.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{607500000\alpha }{78125\times 6^{5}})$$

Calculate 6 to the power of 5 and get 7776.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{607500000\alpha }{78125\times 7776})$$

Multiply 78125 and 7776 to get 607500000.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\frac{607500000\alpha }{607500000})$$

Cancel out 607500000 and 607500000.

$$\frac{\mathrm{d}}{\mathrm{d}\alpha }(\alpha )$$

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

$$\alpha ^{1-1}$$

Subtract 1 from 1.

$$\alpha ^{0}$$

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

$$1$$