derivative of 1/(sqrt(64+5t^2)) — step-by-step solution

To ease differentiation, the expression

\frac{1}{\sqrt{64 + 5 t^{2}}}

is turned into

{(64 + 5 t^{2})}^{- \frac{1}{2}}

Let's differentiate:

\frac{d}{d t} ({(64 + 5 t^{2})}^{- \frac{1}{2}})

Apply the differentiation rule:

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

with f =

64 + 5 t^{2}

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d t} ({(64 + 5 t^{2})}^{- \frac{1}{2}})

- \frac{1}{2} \cdot \frac{d}{d t} (64 + 5 t^{2}) \cdot {(64 + 5 t^{2})}^{- \frac{3}{2}}

- \frac{1}{2} \cdot \frac{d}{d t} (64 + 5 t^{2}) \cdot {(64 + 5 t^{2})}^{- \frac{3}{2}}

ⓘ Multiplying a fraction by a term multiplies its numerator:

- \frac{1}{2} \cdot \frac{d}{d t} (64 + 5 t^{2}) \cdot {(64 + 5 t^{2})}^{- \frac{3}{2}}

- \frac{1 \cdot \frac{d}{d t} (64 + 5 t^{2}) \cdot {(64 + 5 t^{2})}^{- \frac{3}{2}}}{2}

- \frac{\frac{d}{d t} (64 + 5 t^{2}) \cdot {(64 + 5 t^{2})}^{- \frac{3}{2}}}{2}

■The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

- \frac{\frac{d}{d t} (64 + 5 t^{2}) \cdot {(64 + 5 t^{2})}^{- \frac{3}{2}}}{2}

- \frac{\frac{d}{d t} (64 + 5 t^{2})}{2 \cdot {(64 + 5 t^{2})}^{\frac{3}{2}}}

- \frac{\frac{d}{d t} (64 + 5 t^{2})}{2 \cdot {(64 + 5 t^{2})}^{\frac{3}{2}}}

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

\frac{d}{d t} (64 + 5 t^{2})

0 + \frac{d}{d t} (5 t^{2})

- \frac{0 + \frac{d}{d t} (5 t^{2})}{2 \cdot {(64 + 5 t^{2})}^{\frac{3}{2}}}

ⓘ Adding 0 changes nothing:

0 + \frac{d}{d t} (5 t^{2})

\frac{d}{d t} (5 t^{2})

■A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d t} (5 t^{2})

5 \cdot \frac{d}{d t} (t^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d t} (t^{2})

2 t

- \frac{5 \cdot 2 t}{2 \cdot {(64 + 5 t^{2})}^{\frac{3}{2}}}

ⅰ

5 \cdot 2

10

- \frac{10 t}{2 \cdot {(64 + 5 t^{2})}^{\frac{3}{2}}}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

- \frac{10 t}{2 \cdot {(64 + 5 t^{2})}^{\frac{3}{2}}}

- \frac{5 t}{{(64 + 5 t^{2})}^{\frac{3}{2}}}

- \frac{5 t}{{(64 + 5 t^{2})}^{\frac{3}{2}}}

▶Derivative:

- \frac{5 t}{{(5 t^{2} + 64)}^{\frac{3}{2}}}