Solve (81x^12)^1.25

$$( 81 x ^ { 12 } ) ^ { 1.25 }$$

Evaluate

$$243\left(|x|\right)^{15}$$

Solution Steps

Expand \left(81x^{12}\right)^{1.25}.

$$81^{1.25}\left(x^{12}\right)^{1.25}$$

To raise a power to another power, multiply the exponents. Multiply 12 and 1.25 to get 15.

$$81^{1.25}x^{15}$$

Calculate 81 to the power of 1.25 and get 243.

$$243x^{15}$$

Differentiate w.r.t. x

$$3645|x|x^{13}$$

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).

$$1.25\times \left(81x^{12}\right)^{1.25-1}\frac{\mathrm{d}}{\mathrm{d}x}(81x^{12})$$

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}.

$$1.25\sqrt[4]{81x^{12}}\times 12\times 81x^{12-1}$$

Simplify.

$$1215x^{11}\sqrt[4]{81x^{12}}$$