Solve lambda/10*lambda/10

$$\frac { \lambda } { 10 } \cdot \frac { \lambda } { 10 }$$

Evaluate

$$\frac{\lambda ^{2}}{100}$$

Solution Steps

Multiply \frac{\lambda }{10} and \frac{\lambda }{10} to get \left(\frac{\lambda }{10}\right)^{2}.

$$\left(\frac{\lambda }{10}\right)^{2}$$

To raise \frac{\lambda }{10} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\lambda ^{2}}{10^{2}}$$

Calculate 10 to the power of 2 and get 100.

$$\frac{\lambda ^{2}}{100}$$

Differentiate w.r.t. λ

$$\frac{\lambda }{50}$$

Steps Using Derivative Rule for Product

For any two differentiable functions, the derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

$$\frac{1}{10}\lambda ^{1}\frac{\mathrm{d}}{\mathrm{d}\lambda }(\frac{1}{10}\lambda ^{1})+\frac{1}{10}\lambda ^{1}\frac{\mathrm{d}}{\mathrm{d}\lambda }(\frac{1}{10}\lambda ^{1})$$

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}.

$$\frac{1}{10}\lambda ^{1}\times \frac{1}{10}\lambda ^{1-1}+\frac{1}{10}\lambda ^{1}\times \frac{1}{10}\lambda ^{1-1}$$

Simplify.

$$\frac{1}{10}\lambda ^{1}\times \frac{1}{10}\lambda ^{0}+\frac{1}{10}\lambda ^{1}\times \frac{1}{10}\lambda ^{0}$$

To multiply powers of the same base, add their exponents.

$$\frac{1}{10}\times \frac{1}{10}\lambda ^{1}+\frac{1}{10}\times \frac{1}{10}\lambda ^{1}$$

Simplify.

$$\frac{1}{100}\lambda ^{1}+\frac{1}{100}\lambda ^{1}$$

Combine like terms.

$$\frac{1+1}{100}\lambda ^{1}$$

Add \frac{1}{100} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\frac{1}{50}\lambda ^{1}$$

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

$$\frac{1}{50}\lambda$$