Solve (32t^2)^frac{6{5}}

$$( 32 t ^ { 2 } ) ^ { \frac { 6 } { 5 } }$$

Evaluate

$$64t^{\frac{12}{5}}$$

Solution Steps

Expand \left(32t^{2}\right)^{\frac{6}{5}}.

$$32^{\frac{6}{5}}\left(t^{2}\right)^{\frac{6}{5}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{6}{5} to get \frac{12}{5}.

$$32^{\frac{6}{5}}t^{\frac{12}{5}}$$

Calculate 32 to the power of \frac{6}{5} and get 64.

$$64t^{\frac{12}{5}}$$

Differentiate w.r.t. t

$$\frac{768t^{\frac{7}{5}}}{5}$$

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).

$$\frac{6}{5}\times \left(32t^{2}\right)^{\frac{6}{5}-1}\frac{\mathrm{d}}{\mathrm{d}t}(32t^{2})$$

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}.

$$\frac{6}{5}\sqrt[5]{32t^{2}}\times 2\times 32t^{2-1}$$

Simplify.

$$\frac{384}{5}t^{1}\sqrt[5]{32t^{2}}$$

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

$$\frac{384}{5}t\sqrt[5]{32t^{2}}$$