Solve f^7*f^-7/f^2

$$\frac { f ^ { 7 } \cdot f ^ { - 7 } } { f ^ { 2 } }$$

Differentiate w.r.t. f

$$-\frac{2}{f^{3}}$$

Steps Using Chain Rule

Multiply f^{7} and f^{-7} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{1}{f^{2}})$$

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).

$$-\left(f^{2}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}f}(f^{2})$$

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}.

$$-\left(f^{2}\right)^{-2}\times 2f^{2-1}$$

Simplify.

$$-2f^{1}\left(f^{2}\right)^{-2}$$

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

$$-2f\left(f^{2}\right)^{-2}$$

Evaluate

$$\frac{1}{f^{2}}$$

Solution Steps

Multiply f^{7} and f^{-7} to get 1.

$$\frac{1}{f^{2}}$$