derivative of f(x)=a^4+cos^4(x) — step-by-step solution

We save the definition:

f (x) = a^{4} + {\cos (x)}^{4}

Let's perform the differentiation:

\frac{d}{d x} (a^{4} + {\cos (x)}^{4})

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

\frac{d}{d x} (a^{4} + {\cos (x)}^{4})

\frac{d}{d x} (a^{4}) + \frac{d}{d x} ({\cos (x)}^{4})

\frac{d}{d x} (a^{4}) + \frac{d}{d x} ({\cos (x)}^{4})

We differentiate using:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

(where f =

\cos (x)

)

ⓘ The derivative of a constant is zero:

\frac{d}{d x} (a^{4})

0

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({\cos (x)}^{4})

4 \cdot \frac{d}{d x} (\cos (x)) \cdot {\cos (x)}^{3}

0 + 4 \cdot \frac{d}{d x} (\cos (x)) \cdot {\cos (x)}^{3}

ⓘ Adding 0 changes nothing:

0 + 4 \cdot \frac{d}{d x} (\cos (x)) \cdot {\cos (x)}^{3}

4 \cdot \frac{d}{d x} (\cos (x)) \cdot {\cos (x)}^{3}

■The derivative of cos(x) is -sin(x):

\frac{d}{d x} (\cos (x))

- \sin (x)

4 \cdot (- \sin (x)) \cdot {\cos (x)}^{3}

ⓘ The minus sign is placed in front of the product:

4 \cdot (- \sin (x))

-4 \cdot \sin (x)

- 4 \cdot \sin (x) \cdot {\cos (x)}^{3}

▶Derivative:

- 4 \cdot {\cos (x)}^{3} \cdot \sin (x)