Solve d/dxsec5x

$$\frac { d } { d x } \sec 5 x$$

Evaluate

$$\frac{5\tan(5x)}{\cos(5x)}$$

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).

$$\sec(5x^{1})\tan(5x^{1})\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1})$$

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}.

$$\sec(5x^{1})\tan(5x^{1})\times 5x^{1-1}$$

Simplify.

$$5\sec(5x^{1})\tan(5x^{1})$$

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

$$5\sec(5x)\tan(5x)$$

Differentiate w.r.t. x

$$\frac{25\left(2\left(\sin(5x)\right)^{2}+\left(\cos(5x)\right)^{2}\right)}{\left(\cos(5x)\right)^{3}}$$