Solve 5^-2w^0/y^-10

$$\frac { 5 ^ { - 2 } w ^ { 0 } } { y ^ { - 10 } }$$

Evaluate

$$\frac{y^{10}}{25}$$

Solution Steps

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\frac{1}{25}w^{0}}{y^{-10}}$$

Calculate w to the power of 0 and get 1.

$$\frac{\frac{1}{25}\times 1}{y^{-10}}$$

Multiply \frac{1}{25} and 1 to get \frac{1}{25}.

$$\frac{\frac{1}{25}}{y^{-10}}$$

Express \frac{\frac{1}{25}}{y^{-10}} as a single fraction.

$$\frac{1}{25y^{-10}}$$

Differentiate w.r.t. y

$$\frac{2y^{9}}{5}$$

Steps Using Chain Rule

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{25}w^{0}}{y^{-10}})$$

Calculate w to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{25}\times 1}{y^{-10}})$$

Multiply \frac{1}{25} and 1 to get \frac{1}{25}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{25}}{y^{-10}})$$

Express \frac{\frac{1}{25}}{y^{-10}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{25y^{-10}})$$

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$-\left(25y^{-10}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}y}(25y^{-10})$$

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}.

$$-\left(25y^{-10}\right)^{-2}\left(-10\right)\times 25y^{-10-1}$$

Simplify.

$$250y^{-11}\times \left(25y^{-10}\right)^{-2}$$