Solve 4*(-1/2)^3x(-1/2)^2

$$4 \cdot ( - \frac { 1 } { 2 } ) ^ { 3 } x ( - \frac { 1 } { 2 } ) ^ { 2 }$$

Evaluate

$$-\frac{x}{8}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$4\left(-\frac{1}{2}\right)^{5}x$$

Calculate -\frac{1}{2} to the power of 5 and get -\frac{1}{32}.

$$4\left(-\frac{1}{32}\right)x$$

Express 4\left(-\frac{1}{32}\right) as a single fraction.

$$\frac{4\left(-1\right)}{32}x$$

Multiply 4 and -1 to get -4.

$$\frac{-4}{32}x$$

Reduce the fraction \frac{-4}{32} to lowest terms by extracting and canceling out 4.

$$-\frac{1}{8}x$$

Differentiate w.r.t. x

$$-\frac{1}{8} = -0.125$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4\left(-\frac{1}{2}\right)^{5}x)$$

Calculate -\frac{1}{2} to the power of 5 and get -\frac{1}{32}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4\left(-\frac{1}{32}\right)x)$$

Express 4\left(-\frac{1}{32}\right) as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4\left(-1\right)}{32}x)$$

Multiply 4 and -1 to get -4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-4}{32}x)$$

Reduce the fraction \frac{-4}{32} to lowest terms by extracting and canceling out 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\frac{1}{8}x)$$

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

$$-\frac{1}{8}x^{1-1}$$

Subtract 1 from 1.

$$-\frac{1}{8}x^{0}$$

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

$$-\frac{1}{8}$$