Solve c[(32)^3/2]^-2/5

$$c [ ( 32 ) ^ { 3 / 2 } ] ^ { - 2 / 5 }$$

Evaluate

$$\frac{c}{8}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and -\frac{2}{5} to get -\frac{3}{5}.

$$c\times 32^{-\frac{3}{5}}$$

Calculate 32 to the power of -\frac{3}{5} and get \frac{1}{8}.

$$c\times \frac{1}{8}$$

Differentiate w.r.t. c

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

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and -\frac{2}{5} to get -\frac{3}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c\times 32^{-\frac{3}{5}})$$

Calculate 32 to the power of -\frac{3}{5} and get \frac{1}{8}.

$$\frac{\mathrm{d}}{\mathrm{d}c}(c\times \frac{1}{8})$$

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

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

Subtract 1 from 1.

$$\frac{1}{8}c^{0}$$

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

$$\frac{1}{8}\times 1$$

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

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