Solve θ(0.001)1/3

$$\theta ( 0.001 ) \frac { 1 } { 3 }$$

Evaluate

$$\frac{\theta }{3000}$$

Solution Steps

Convert decimal number 0.001 to fraction \frac{1}{1000}.

$$\theta \times \frac{1}{1000}\times \frac{1}{3}$$

Multiply \frac{1}{1000} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\theta \times \frac{1\times 1}{1000\times 3}$$

Do the multiplications in the fraction \frac{1\times 1}{1000\times 3}.

$$\theta \times \frac{1}{3000}$$

Differentiate w.r.t. θ

$$\frac{1}{3000} = 0.0003333333333333333$$

Steps Using Definition of a Derivative

Convert decimal number 0.001 to fraction \frac{1}{1000}.

$$\frac{\mathrm{d}}{\mathrm{d}\theta }(\theta \times \frac{1}{1000}\times \frac{1}{3})$$

Multiply \frac{1}{1000} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}\theta }(\theta \times \frac{1\times 1}{1000\times 3})$$

Do the multiplications in the fraction \frac{1\times 1}{1000\times 3}.

$$\frac{\mathrm{d}}{\mathrm{d}\theta }(\theta \times \frac{1}{3000})$$

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

$$\frac{1}{3000}\theta ^{1-1}$$

Subtract 1 from 1.

$$\frac{1}{3000}\theta ^{0}$$

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

$$\frac{1}{3000}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{3000}$$