derivative of sec(x^5+x) — step-by-step solution

Let's differentiate:

\frac{d}{d x} (\sec (x^{5} + x))

We differentiate using:

\frac{d}{d x} (\sec (f)) = \frac{d}{d x} (f) \cdot \sec (f) \cdot \tan (f)

(where f =

x^{5} + x

)

ⓘ The derivative of sec(x) is sec(x)*tan(x):

\frac{d}{d x} (\sec (x^{5} + x))

\frac{d}{d x} (x^{5} + x) \cdot \sec (x^{5} + x) \cdot \tan (x^{5} + x)

\frac{d}{d x} (x^{5} + x) \cdot \sec (x^{5} + x) \cdot \tan (x^{5} + x)

ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{5} + x)

\frac{d}{d x} (x^{5}) + 1

(\frac{d}{d x} (x^{5}) + 1) \cdot \sec (x^{5} + x) \cdot \tan (x^{5} + x)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{5})

5 x^{4}

(5 x^{4} + 1) \cdot \sec (x^{5} + x) \cdot \tan (x^{5} + x)

▶Derivative:

\sec (x^{5} + x) \cdot \tan (x^{5} + x) \cdot (5 x^{4} + 1)