Solve (c^frac{3{4}})^7

$$( c ^ { \frac { 3 } { 4 } } ) ^ { 7 }$$

Differentiate w.r.t. c

$$\frac{21c^{\frac{17}{4}}}{4}$$

Steps Using Chain Rule

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).

$$7\left(c^{\frac{3}{4}}\right)^{7-1}\frac{\mathrm{d}}{\mathrm{d}c}(c^{\frac{3}{4}})$$

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}.

$$7\left(c^{\frac{3}{4}}\right)^{6}\times \frac{3}{4}c^{\frac{3}{4}-1}$$

Simplify.

$$\frac{21}{4}c^{-\frac{1}{4}}\left(c^{\frac{3}{4}}\right)^{6}$$

Evaluate

$$c^{\frac{21}{4}}$$