Question Statement
Find the derivative of the function:
y=(x3−10)910
with respect to x.
Background and Explanation
This problem involves differentiating a composite function using the chain rule. Before applying the chain rule, we rewrite the rational expression using negative exponents to make the power rule applicable.
Solution
To differentiate this function, we first rewrite it using negative exponents to simplify the application of the power rule and chain rule.
Step 1: Rewrite the function
y=(x3−10)910=10(x3−10)−9
Step 2: Apply the chain rule
We treat this as a composite function where the outer function is 10u−9 and the inner function is u=x3−10. Using the chain rule dxdy=dudy⋅dxdu:
dxdy=10⋅(−9)(x3−10)−9−1⋅dxd(x3−10)
Step 3: Simplify the exponents and differentiate the inner function
The derivative of the inner function (x3−10) with respect to x is 3x2:
dxdy=−90(x3−10)−10⋅(3x2−0)
Step 4: Multiply and simplify
dxdy=−90(x3−10)−10⋅3x2=−270x2(x3−10)−10
Step 5: Convert back to positive exponent form
dxdy=(x3−10)10−270x2
- Negative Exponent Rule: un1=u−n (used to rewrite the function before differentiation)
- Chain Rule: dxd[f(g(x))]=f′(g(x))⋅g′(x) (applied to differentiate the composite function)
- Power Rule: dxd[un]=n⋅un−1⋅dxdu (used on the outer function)
- Power Rule for Polynomials: dxd[xn]=nxn−1 (used to differentiate the inner function x3−10)
Summary of Steps
- Rewrite the rational function using negative exponents: y=10(x3−10)−9
- Apply the chain rule: differentiate the outer function 10u−9 to get −90u−10, keeping the inner function unchanged
- Multiply by the derivative of the inner function (x3−10), which is 3x2
- Simplify the numerical coefficients: −90×3=−270
- Convert the final answer back to fractional form with positive exponents in the denominator