Solve (49x^3)^-frac{3{2}}

$$( 49 x ^ { 3 } ) ^ { - \frac { 3 } { 2 } }$$

Evaluate

$$\frac{1}{343x^{\frac{9}{2}}}$$

Solution Steps

Expand \left(49x^{3}\right)^{-\frac{3}{2}}.

$$49^{-\frac{3}{2}}\left(x^{3}\right)^{-\frac{3}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and -\frac{3}{2} to get -\frac{9}{2}.

$$49^{-\frac{3}{2}}x^{-\frac{9}{2}}$$

Calculate 49 to the power of -\frac{3}{2} and get \frac{1}{343}.

$$\frac{1}{343}x^{-\frac{9}{2}}$$

Differentiate w.r.t. x

$$-\frac{9}{686x^{\frac{11}{2}}}$$

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).

$$-\frac{3}{2}\times \left(49x^{3}\right)^{-\frac{3}{2}-1}\frac{\mathrm{d}}{\mathrm{d}x}(49x^{3})$$

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}.

$$-\frac{3}{2}\times \left(49x^{3}\right)^{-\frac{5}{2}}\times 3\times 49x^{3-1}$$

Simplify.

$$-\frac{441}{2}x^{2}\times \left(49x^{3}\right)^{-\frac{5}{2}}$$